1 Sequences and Series

Transcription

1 Sequences and Series. Introduction What is a sequence? What is a series? Very simply a sequence is an infinite list of things. Normally we consider now only lists of numbers - so thus, the list of all natural numbers, 2, 3, 4, is an example. Another example is the list and a third is, 2, 3, 4,,,,,,, Now briefly, what is a series? Given any sequence such as the three above, simply replace the commas with plus signs. The result is a series, an expression which in some cases represents a particular number and in other cases may have no particular meaning. We can call such an expression an infinite sum. However, this is a rather odd use of language, since we can only add members of one number at a time, and it would take an infinite amount of time to add infinitely many numbers. This is one of the problems first noted by the ancient Greeks and forms the foundation of what is known as Zeno s paradox. In what follows we will elaborate on these ideas..2 Sequences So - a sequence is a list of numbers. This means there is a first, a second, a third, and so on, and if n is an arbitrary natural number we can consider the n th number in the list. To define a particular sequence we need to present a rule that describes which number gets assigned to the n th position. If we abstractly let a n stand for the n th number, for the three examples above, we can describe the rules which define the sequences. The sequence,, 2, 3, 4, has as its rule a n = n. The sequence, 2, 3, 4, has the rule a n = n. The third sequence,,,,,, has a more complicated rule, namely a n = if n is an odd number - that is the first, third, fifth, elements of the sequence and a n =, if n is an even number. More compactly, we could rephrase this rule as: a n = ( ) n+.

2 Substitute various values of n to see that this works. An abstract sequence - one in which we have not yet attributed any values to the various positions in the sequence - is denoted as {a n } which in turn is short hand for the set notation {a n : n =, 2, 3, }..2. The Fibonacci Sequence There is a famous example which pops up as a model of various structures in the real world. This is the so called Fibonacci Sequence which was first mentioned by Leonardo Fibonacci, A.D., one of the first european mathematicians. Fibonacci is famous for having been the first among the europeans to use our current system of writing numbers, the Hindu-Arabic numeral system. Fibonacci came upon the Fibonacci sequence after examining the breeding patterns of his friend s rabbit farm. The male rabbits were sold in the market but the females were kept for breeding, and hence were the only ones he was interested in keeping track of. He observed that a female will on average produce a single female offspring in one month and after another month the offspring is mature enough to become pregnant. The friend started with female rabbit. After month there are then two, the mother and her new born daughter. After the second month the first rabbit gives birth again so there are now three rabbits, the mother, the daughter and a second daughter. The first daughter is now a month old and to produce a granddaughter after a month. In other words now after the thirds month all those females present after the first month, namely the mother and the first daughter, can produce off spring. Thus the population after the third month equals the population after the first plus all those present after the second. Similarly all those after the second month can produce for the fourth month. The population after the fourth month is then all those produced by those alive in the second month, namely the total number in the second month, plus those existing in the third month. In general, if f n denotes the population after the n th month, then the population after n + months is the the sum of those producing after n months plus those present after n months. In other words for n =, 2, 3, 4, after n months there will be female rabbits. The numbers are f n+ = f n + f n, 2, 3, 5, 8, 3, 2, 34, 55, 89, 2

3 .2.2 Arithmetic Sequences The following is an example of an arithmetic sequence 3, 7,, 5, 9, All arithmetic sequences are similar. They depend on two numbers: the starting point c ( c = 3 in the example and the common difference d (d = 4 in the example) that is successively added to the previous term. Thus = 7, =, + 4 = 5 and so on. In general now, if {a n } is an arithmetic sequence determined by starting point c and common difference d, then the terms are: a, a + d, a + 2d, a + 3d, where for each n, a n+ = a n + d. From the pattern we can immediately see that for each n, the n th term of the sequence is {a + (n )d}.2.3 Limits Some sequences have the property that the terms of the sequence get closer and closer to some specific fixed number called the limit of the sequence. If this happens, the sequence is said to converge. For the examples in which a n = n the sequence evidently has a limit equal to 0. The sequence with a n = for n odd and a n = for n even bounces between the two values and and hence has no limit. The fibonacci sequence is one in which the terms keep increasing without any bound and hence also there is no limit. The task of determining wether or not a sequence converges and to what limit is of immense importance in mathematics and it is not necessarily an easy task. Consequently we have had to be very careful to specify precisely what is meant by a limit. What does it mean for terms to get closer and closer to some number? We need to have a good understanding of this. One way of rephrasing might look like A sequence converges to a limitl provided there is some position in the sequence beyond which all terms of the sequence are arbitrarily close to L. Seems to make sense - we can rephrase again as: A sequence {a n } converges to a limit L provided there is a natural number N representing the N th position in the sequence with the property that if n is a natural number greater than N then a n is arbitrarily close to L 3

4 The difficulty with this is that the words arbitrarily close are still too vague to be useful. We need to get a handle on this idea. What we mean is that for any arbitrary small number representing a measure of closeness, the distance between a n and L must be less than this number. Traditionally this arbitrary measure of closeness is denoted with the Greek letter epsilon ɛ. We are now in the position saying something quite precise. A sequence {a n } converges to a limit L provided for every measure of closeness ɛ, there is a natural number N such that for every natural number n if n > N then the distance between a n and L is less than ɛ Since distance between points on the real line is best measured using the absolute value notation, we can rephrase again as follows. Definition Limit of a Sequence A sequence {a n } converges to a limit L provided for every measure of closeness ɛ > 0, there is a natural number N such that for every natural number n, n > N a n L < ɛ. If this is the case, we say that the limit of a n as n goes to infinity is L. This is abbreviated by writing lim n a n = L More succinctly we can express this symbolically as follows ɛ > 0, N N such that n N, n > N a n L < ɛ Lets see how this definition might be applied. Example Consider the sequence {a n } where a n = 2n 2 n. And lets use the definition above to prove that the sequence has a limit L =. Following the definition, we first need to pick an arbitrary small number ɛ > 0, and then somehow find a suitable integern, which of course must be dependent in some way on ɛ. First note that 2 n 2 n = 2 n. Therefore, since the numbers 2 keep getting smaller, the numbers a n n = 2n 2 keep getting n larger, but remain always less than. Pick an integer N sufficiently large so that N < ɛ, and note that for n > N, we then have n < ɛ. Now lets see if we can prove that a n < ɛ. If we can do this, then according to the definition, we have also then proved that the sequence converges to. Observe however that if n > N, then a n = 2n 2 n = n = 2 2 n = 2 n < n < ɛ, 4

5 since by substituting some values for n, it becomes clear that the inequality 2 n < n is indeed true. Thus we have shown that for n > N, a n < ɛ. Following is an example where it appears that the terms of the sequence approach a limit, but it is not immediately clear what this limit is. Example Consider the following sequence where a n = (+ n )n. The first 6 terms are: or with decimal approximations ( 3 ) 2, ( 4 ) 3, ( 5 ) 4, ( 6 ) 5, ( 7 6 2, ) 2, 2.25, , 2.444, , It appears that the terms are increasing very slowly. Lets check a term for a large value of n - say n = 000. Using a computer and displaying only the first 4 decimals, we see that a 000 = ( ) So it is also reasonable to assume that a limit, if it exists, is somewhere in the neighborhood of this last figure. It turns out that this limit is an important constant in nature that occurs in many situations. It is given the name e. We will discuss it later in the context of a study of logarithms..2.4 Combining Sequences Suppose we are given two sequences- say: 2, 4, 8, 6, 32, and 3, 9, 27, 8,. We can then form two new sequences by either successively adding a term of the first to a term of the second or successively multiplying a term of the first with a term of the second. In the case of adding, we get the new sequence: and in the case of multiplying, we get 2 + 3, 4 + 9, , 6 + 8,, 2 3, 4 9, 8 27, 6 8,. 5

6 More generally we have the following definitions of what it means to add and to multiply two sequences Definition Given two sequences {a n } and {b n } we define a form of addition and a form of multiplication of sequences. The operations are denoted as and.. The sum of {a n } and {b n } is 2. The product of {a n } and {b n } is {a n } {b n } = {a n + b n }. {a n } {b n } = {a n b n }. In the same way we can define the multiplication of a sequence by a number. Given a sequence {a n } and an arbitrary real number c, we define c {a n } = {ca n }..2.5 The Limit of the Sum is the Sum of the Limits What we need to do next is to show that if we are given two sequences {a n } and {b n } each of which has a limit - say lim n a n = L and lim n b n = M - then the limit of the sum of the two sequences is the sum L + M. Symbolically we need to show that lim a n + b n = L + M. n In order to prove this result the only tools we have are the definition of a limit. To do this is a bit of a stretch, but it is well worth the effort. The definition has complications that puzzle most on first acquaintance. Working through the proof will help straighten things out. Basically this is an exercise in careful reading of the definition. Proof : We need to show that for every arbitrary but fixed measure of closeness denoted by a small positive number ɛ, we need to find a positive integer N representing some far distant point in the sequence, so that for each integer n, if n > N, then lim n a n +b n (L+M) < ɛ. All we know is that lim n a n = L and lim n b n = M. Since we have the arbitrary ɛ, lets consider the smaller number ɛ/2 and use it when considering the limits L and M. We have then associated with these limits two positive integers N L and N M with the property that for n > N L and n > N M. a n L < ɛ/2 b n M < ɛ/2 6

7 If we let N be the larger of N L and N M, then for n > N both of the above inequalities are still true. All we need to do now is to add the above two lines. We get for n > N, a n L + b n M < ɛ/2 + ɛ/2 = ɛ. Now we use the fact regarding absolute values that the absolute value of a sum of numbers must be less than or equal to the sum of the absolute values- i.e. for any two numbers x and y, x + y < x + y. Applying this we see that (a n + b n ) (L + M) = (a n L) + (b n M) < a n L + b n M. Combining with the previous line we have what we need - namely that for n > N, a n + b n (L + M) < ɛ.2.6 An obvious (?) result Suppose we have a sequence {a n } If we set b n = a n+, we get a new sequence a, a 2 ; a 3, a 2, a 3, a 4,, whose only difference is that the first term is omitted. I am hoping that it appears obvious that if the first sequence {a n } has a limit, then the second {b n } has the same limit. A simple proof using the formalism of the definition is easy to write down..3 Series As mentioned in the introduction, given a sequence a, a 2, a 3,, the corresponding series is the expression a + a 2 + a 3 +. We also mentioned that in sometimes this expression has no mathematical meaning. And indeed, what do we mean by an infinite sum? This question needs to be answered. The answer has not been obvious, and has represented a mental tangle for a few thousand years. Fortunately the answer is not difficulty to grasp. We start with the notion of a partial sum. 7

8 Given a sequence {a n } and a natural numbern, the N th partial sum is simply the sum of the first N terms a + a 2 + a a N + a N. There is a simplified way of representing the above using the so-called sigma notation. The Greek letter capital sigma, is used in the same way as our letter S, and is used as an abbreviation for the word Sum. The N th partial sum of a sequence a n is then written This can be translated as, form the sum of the sequence {a n } from k = to k = N. Example : If a n = ( 3 5 )n and n = 4 then N k= a k 4 k= a k = 4 k= ( 3 5 )n = ( 3 5 ) + ( 3 5 )2 + ( 3 5 )3 ( 3 5 )4 = = =.3056 Using the concept of partial sums, we can now say what we mean by an infinite sum. Using the sigma notation for a sequence {a n } the expression a, a 2, a 3,, can be written as k= a k. We can also now say what we mean by these expressions - that is: what is meant by an infinite sum? Given a sequence {a n }, lets denote the n th partial sum by S n. For the various values of n this gives us a sequence, S, S 2, S 3,. We then say that the expression k= a n makes sense only when the limit of the partial sums exists. If this is the case, we say that the series converges. In other words there is a number S such that S = lim n S n exists. Definition : Given a sequence {a n }, the series k= a k converges if the limit of the partial sums S = lim n S n exists. Many of the functions that arise naturally in mathematics do not have an easy formulation, and sometimes the only way to get a good handle on them is to find a series approximation. As a result the task of determining when a particular series converges is important. Straight from the definition of convergence we can however derive a good first guess. Here s how it goes. Suppose we have a convergent series lim n a n = L. Let {S n } be the sequence of partial sums with limit L. If we set T n = S n+, we have a new sequence {T n } of partial sums which differs from {S n } only in the first term and thus has the same limit 8

9 L. Now if we examine the limit of the sequence {a n } of terms of the series, we see that since n+ n T n S n = S n+ S n = a k a k = a n+ k= k= lim n a n+ = lim n T n S n = lim n T n + ( )S n = lim n T n + lim n ( )S n = lim n T n + ( ) lim n S n = L L = 0 We can state the result compactly as follows. Theorem : Given the series k= a n, if k= a n converges, then lim n a n = 0. The result becomes more useful and interesting if we phrase it as the contrapositive of the above. ( Remember, given an implication p q, the contrapositive is the statement q p and has precisely the same truth values as the original p q.) The contrapositive of the theorem is: Given the series k= a n, if lim n a n 0, then k= a n does not converge. You can see now that this provides a very simple initial test. If the limit of the sequence of terms of the series is not zero, then the series cannot converge. For instance suppose we have the series,,,,, which alternates between + and and can be expressed in sigma notation as k= ( )k+. Observing that the limit of the terms does not converge at all, it certainly does not converge to zero. By our result the corresponding series also cannot converge. Thus in this case the notion of the infinite sum is meaningless. Example : For a simple example consider the arithmetic sequence, 2, 5, 8,, Since the limit of the sequence is clearly not zero, we know that if we put plus signs in place of commas, the corresponding series cannot converge. In particular calculating a few terms of the sequence of partial sums, we have S = 2, S 2 = 7, S 3 = 5, S 4 = 26, S 5 = 40, S 6 = 57, which also clearly does not converge. 9

10 .3. Partial sums of arithmetic sequences Here is a problem. Imagine an auditorium in the traditional wedge shape - narrow near the stage fanning out towards the rear. Suppose the first row contains 20 seats and that as one proceeds to the back of the hall each row contains 3 more seats than the previous row. Further suppose there are 35 rows. The question then is, without any counting, is there a way of quickly determining the total number of seats? Lets simplify the problem by considering only 4 rows. Perhaps there is some pattern that we can generalize. The first thing to realize is that we have an arithmetic sequence on our hands in which the initial term a equals 20 and the common difference d equals 3. The sequence is: 20, 23, 26, 29, and the corresponding partial sum is S 4 = The trick now is to write down the terms of S 4 in reverse order and then add the two expressions and modestly rewrite the sum. We get: S 4 = S 4 = S 4 = ( ) + ( ) + ( ) + ( ) In other words, 2S 4 = 4( ) or S 4 = 4 2 ( ). I think it is clear that this that this pattern will hold no matter how rows are to be considered. Suppose now that there are n rows. The sum then becomes ( ) S n = (n )3, and going through the same procedure as above we get 2S n = n( (n )3) so that S n = n (2(20) + (n )3). 2 For n = 35, we have S n = 35 2 (40 + (34)3) = 2485 For an arbitrary arithmetic sequence a, a + d, a + 2d,, where the common difference is d, and the n th term is a n = a + (n )d, we can use exactly the same method for finding S n. In particular we see that S n = n 2 (a + a n ) = n 2 (a + a + (n )d) = n (2a + (n )d). 2 0

11 .3.2 Geometric series Consider the situation of a ball that has been dropped from a height of 6 feet with the property that the first bounce is 8 feet high and that each subsequent bounce is half the height of the previous. The question is what is the total distance that the ball covers counting each of the bounces both up and down. Lets compute what happens after a few bounces - 6 feet down + 8 feet up + 8 feet down + 4 feet up + 4 feet down + 2 feet up + 2 feet down +. Evidently we have a series that looks like 6 + ( ) which can also be written after factoring 6 out of the parenthesis as Notice we have decided to start the summation at k = 0 instead of k =. So what we have is 6 plus 6 times a series. This will only make sense if the series converges. We also need to know to which number the series converges. This is what we do. Write down the nth partial sums n and subtract from it 2 times S n. k=0 2 k The result is, Solving for S n,we get S n = n 2 S n = n + 2 n+ S n 2 S n = 2 n+ S n = 2 n+ 2 We now need to take the limit of the sequence of partial sums S n. Our calculation above leaves us in a good position, for the only portion that varies is the term, and we know 2 n+ that lim n 2 n+ = 0. Thus lim S n = n 2 Thus the series k=0 = 2, and the bouncing ball problem makes sense, and the answer 2 k is is that the ball travels a total of then 6 + 6(2) = 48 feet before it comes to rest. = 2

12 The series k=0 is an example of what we call a geometric series. If we were to replace 2 k the fraction 2 in the above calculations with a number r everything would work fine and we would get as far as S n = rn+ r. As long as < r <, it similarly follows that lim n r n+ = 0. Under these conditions we then know that the series k=0 rk = r. Lets look at another example of how this might work. Example : Consider the series k=0 ( 3 5 )k. Does it converge? If so, to what value? To answer this we need to put it in the form above. All we need to do is to set r = 3 5. From our work we then know that the infinite sum k=0 ( 3 5 )k has value = Zeno s race course paradox Pythagoras was born on the Greek island of Samos in about 560 B.C. As a young man he traveled to Egypt and spent 24 years, so the legends say, learning the mysteries of the Egyptian temples. When Egypt was conquered by the Persians he traveled to Babylon where, again as the legend says, he studied the knowledge of the Babylonian astronomers. Then after 36 years he returned to the Greek world and set up a school of philosophy in the Greek city state of Crotona on the Italian peninsula. One of the tenets of his philosophy was that all reality was composed in various configurations and quantities of a basic element called the monad. Further he taught that real understanding of the workings of the universe could be comprehended by studying the various parts, breaking the parts into sub-parts and studying them, and so on until one is at the level of the monad - a sort of a destructive analysis. Using a point as a representative of the monad, Pythagoras also considered the reverse procedure of analyzing fundamental patterns arising from geometric combinations and configurations of points. In the near by Greek city of Elea there was a competing school of philosophy led by Parmenides. Much less is known of the teachings of Parmenides. From the scanty evidence that remains, it is apparent however that Parmenides had different ideas regarding the structure of the universe. His position was that the belief that the reality was built up bit by bit from some abstract element was fundamentally wrong. Understanding could not be gained by chopping things into parts. Zeno was a student of Parmenides who set out to demonstrate that the Pythagoreans were wrong. To do this he constructed a set of seemingly plausible arguments, now known as the Zeno paradoxes. These arguments resulted in absurd conclusions, and Zeno s assertion was that the errors were the result of assumptions based on pythagorean teachings. The 2

13 race course paradox is one such arguments. It goes like this - imagine a race course of one kilometer in length and a runner who is to run the course at the slow constant speed of kilometer and hour. Using the formula distance = rate x time, he must first cover the first half of the course in /2 hour. He then must cover half of the remaining half of the course, namely /4 of the course, in /4 hour. Next he must cover half of the remaining /4 - namely and /8 of the course, in /8 of and hour. And so on. The total amount of time to complete the course is then the sum of all such time intervals - that is, the sum of infinitely many positive numbers. Zeno asserted that such a sum would necessarily be infinite and consequently the runner could never reach the end of the course. Since this is an absurd conclusion, there must then be something wrong with the assumptions underlying the argument. Zeno s position was that the only possible error was the assumption that the race course could be divided into infinitely many parts - an idea inherent in the pythagorean teachings. Since the teachings result in an error, Zeno then claims that the teachings themselves were in error. Today with our knowledge of series there is of course a different explanation. Zeno has constructed for us the geometric series /2, /4, /8,, /2 n, which we now know converges to the value. The error instead is Zeno s assumption that an infinite sum of positive numbers is necessarily infinite..3.4 The harmonic series Pythagoras is also credited with the creation of the musical scales we use today. He noticed through experimentation that if you take a string which for instance gives a middle C when plucked and then reduce the length by 2/3 rds, the resulting note is the G above middle C. This interval is called a fifth. The two notes when played together emit a natural resonance between them which gives a pleasing sound. On the other hand if the string is divided in half and then plucked, the resulting note is the C above middle C and when the two notes are played together the vibration patterns are nearly identical and the two notes sound as one. The interval between these notes is called an octave. The intervals of a fifth and an octave are fundamental to our notions of harmony and form the basis of all musical compostion. The string lengths are:, 2/3, /2. These numbers are said to be in harmonic progression with 2/3 defined as the harmonic mean of and /2. If we look at the sequence of reciprocals, we get, 3/2, 2. 3

14 These numbers form part of an arithmetic sequence with the common difference being /2. More generally, any sequence is said to be in harmonic progression if the sequence of reciprocals is an arithmetic sequence. The classic example is the following sequence, /2, /3, /4, /n The natural question that now arises is - does the corresponding series /k =, /2, /3, /4, /n k= converge? The answer is no. Lets see why this is true. The trick is to look at some partial sums to see if there is some pattern that will allow us to make a general statement. Look then at the sum S 32 of the first 32 terms. Notice that we can put parentheses around increasingly large groups of terms so that the sums of the elements in parentheses is greater to /2. For instance () + (/2) + (/3 + /4) + (/5 + /6 + /7 + /8) + (/9 + + /6) + (/7 + + /32 In particular we know that for the middle term (/5+/6+/7+/8) each of the fractions other than /8 is greater than /8 and there are four such fractions - thus (/5 + /6 + /7 + /8) > (/8 + /8 + /8 + /8) = 4/8 = /2 Calculating the partial sums, we see that: S 2 = + 2 = 3 2 > 2, and for subsequent sums -say S 4 we have S 4 = () + (/2)(/3 + /4) = S 2 + (/3 + /4) < = 2( 2 ). Similarly S 8 = S 2 3 > 3( 2 ) S 6 = S 2 4 > 4( 2 ) S 32 = S 2 5 > 5( 2 ) Using the same analysis we can say that S 2 n > n( 2 ). Thus, it is evident that the partial sums keep increasing without bound - in other words the series cannot converge. 4