12.1 Arithmetic Progression Geometric Progression General things about sequences

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1 ENGR11 Engineering Mathematics Lecture Notes SMS, Victoria University of Wellington Week Five. 1.1 Arithmetic Progression An arithmetic progression is a sequence where each term is found by adding a fixed quantity, the common difference d, to the previous term. It is given by the general formula a, a + d, a + d, a + 3d,..., a + (k 1)d,... The recursive form for an arithmetic progression is x[k] = x[k 1] + d, which needs to be started with a value for the first term x[0] = a. 1. Geometric Progression A geometric progression is a sequence where each term is found by multiplying a fixed quantity, the common ratio r, by the previous term, for example start with 1 and multiply by 1/ to get the geometric progression 1, 1/, 1/4, 1/8,... The general form is and the recursive form is a, ar, ar, ar 3,... x[k] = r x[k 1], k = 1,, 3,... which needs to be started with a value for x[0]. 1.3 General things about sequences A sequence can be unbounded, that is, it can grow without stopping, like, 4, 6, 8, 10,.... Or it can be bounded, that is, its terms are ited in size, for example 1, 1/, 1/4, 1/8,... whose terms decrease to zero and are bounded by 1 and zero. 58

2 The notation we use for the it of x[k] as k tends to infinity is zero is to write x[k] = 0. k A sequence is said to converge to a it l if, by proceeding far enough along the sequence, all subsequent terms can be made to lie as close to l as we desire, and l is finite. We then write k x[k] = l. A sequence is divergent if it does not converge to any it. The sequence 1, 1, 1, 1, 1,... is bounded, but does not converge to a it. It oscillates. If terms in a sequence increase without bound, it is divergent. Some rules that we can use to help decide if a sequence is convergent are: If x[k] and y[k] are two convergent sequences with and k x[k] = l 1 k y[k] = l then it follows that 1. The sequence x[k] ± y[k] has it l 1 ± l.. The sequence cx[k] has it cl The sequence x[k]y[k] has it l 1 l. 4. The sequence x[k]/y[k] has it l 1 /l provided l We can always assume that k 1 k m = 0 m > 0. Example: find the it of the infinite sequence x[k] = k 1 k + 1, k = 0, 1,, 3,... The key here is to divide top and bottom of the kth term by k, to get ( ) 1 1/k x[k] =. k k 1 + 1/k 59

3 Using the rules above, we can then write x[k] = k (1 1/k) k k (1 + 1/k) rule 4 = k (1) k (1/k) k (1) + k (1/k) = 1 0 rule = 1 rule 1 Note that in the above, we used the obvious fact that the it of the sequence 1, 1, 1, 1,... is Series Croft, section 6.3, p. 199 If terms of a sequence are added together, we get a series. For example, the series S = 1 + 1/ + 1/3 + 1/4 is a finite series. The series S = 1 + 1/ + 1/3 + 1/ is an infinite series. We use the sigma notation for series, so that the sum of the first n terms is written in general n S n = x[k], k=1 and this means the finite sum x[1] + x[] + x[3] x[n] Arithmetic series An arithmetic series is the sum of a (finite) arithmetic progression. For example, S = which adds up to 15. There is a quick way to add up long arithmetic series: add the reversed series to S to get S =

4 that is, to get the correct result S = 15. S = 5 6 = 30 Doing this for the general form gives a nice formula for a general arithmetic series: added to so that S k = a + (a + d) + (a + d) (a + (k )d) + (a + (k 1)d) S k = (a + (k 1)d) + (a + (k )d) (a + d) + (a + d) + a S k = (a + (k 1)d) + (a + (k 1)d) (a + (k 1)d) and there are k of these terms, so that it follows, dividing by on both sides, that the sum of an arithmetic series is k 1 S k = (a + id) = k (a + (k 1)d). i=0 1.5 Sum of a finite geometric series A geometric series is the sum of the terms of a geometric progression, A useful formula for this sum is S k = a + ar + ar ar k 1 = i=k 1 i=0 ar i. S k = a(1 rk ) 1 r, provided r 1 and this formula may be obtained by multiplying S k by r and subtracting the result from S k (Croft, p.01). 1.6 Sum of an infinite geometric series The finite sum of k terms is given above and called S k. We think about the sequence of partial sums S k as k, and ask if it converges in this it. 61

5 Noting that provided r < 1, r k 0 as k, we see that S r = r a(1 0) 1 r = a 1 r. When the sequence of partial sums S k converges, we say that the infinite series also converges, and we see from the above that it converges to the value a/(1 r), so that ar i = i=0 a 1 r, provided r < 1 and if r 1 the series diverges in fact. Geometric Series example Find the sum to k terms of the following series, and then determine the sum to infinity: This is a geometric series, with first term a = and common ratio r = 1/3. Hence S k = a(1 rk ) = ( ) k = 3 (1 13 ) 1 r /3 k Sending k means the term 1 vanishes, that is, 3 k ( ) 1 = 0 k 3 k so that S = The Binomial Theorem (Croft section 6.4, p. 04) This theorem extends the idea of and multiplying this by a + b gives (a + b) = a + ab + b (a + b) 3 = a 3 + 3a b + 3ab + b 3. 6

6 Going to higher powers gets tedious to work out. Along comes Pascal s Triangle to the rescue. (Is it a bird? Is it a plane? No, it s Pascal s Triangle!) Pascal s Triangle: n = 0: 1 n = 1: 1 1 n = : 1 1 n = 3: n = 4: n = 5: n = 6: You can see that each row in Pascal s triangle is formed by adding together terms in the row above. The nth row tells you the coefficients of the terms in (a + b) n. The terms go in increasing powers of a and decreasing powers of b, and the powers alway add to n. For example (a + b) 4 = 1.a 4 + 4a 3 b + 6a b + 4ab b 4. In fact, for large powers even Pascal s Triangle is daunting. Then the binomial theorem, a true superhero, comes to the rescue: The binomial theorem states that, when n is a positive integer, (a + b) n = a n + na n 1 b + n(n 1) a n b +! The extended binomial theorem states that, when x < 1, n(n 1)(n ) a n 3 b b n. 3! (1 + x) n n(n 1) = 1 + nx + x n(n 1)(n ) + x ,! 3! and now it is not necessary that n be a positive integer. n can be negative, or a fraction, in general. This is a very useful theorem in applied mathematics. 63

7 A true superhero the extended binomial theorem. For example, provided x < 1, 1 1 x x x 1 + x = (1 + x) = x +... = and when x is small, we can approximate 1 + x as accurately as we like by taking enough terms. 1 64

8 14 Power Series Croft, section 6.5, p. 08 Superpowers abound: An example of a power series is 1 + x + x + x It is basically an infinite polynomial. Many functions have well-known power series expansions, for example if x is in radians, sin x = x x3 3! + x5 5! x7 7! +... This series converges for any value of x. For example, if x = 0.5, then sin(0.5) = ! 5! and summing just the first three terms gives the approximation sin(0.5) which may be compared with the value that is accurate to seven significant figures sin(0.5) A power series may sometimes converge only for x small enough, say x < R, that is, R < x < R. If R is the largest such value, giving convergence of the power series, we way that R is the radius of convergence of the power series. Do you know why the word radius is used here? 65

9 The open interval ( R, R) is called the interval of convergence. Here are two more well-known power series, with infinite radii of convergence: cos x = 1 x x4 x ! 4! 6! ex = 1 + x + x in radians x x3 x ! 3! 4! 66