Series Professor Richard Blecksmith richard@math.niu.edu Dept. of Mathematical Sciences Northern Illinois University http://math.niu.edu/ richard/math230 These notes are taken from Calculus Vol I, by Tom M. Apostol, 966.. Infinite Series From a given sequence of real numbers a,a 2,a 3,...,a n,... we can always generate a new sequence by adding together successive terms. We may form, in succession, the partial sums S = a, S 2 = a +a 2, S 3 = a a +a 2 +a 3, and so on. The partial sum of the first n terms is S n = a +a 2 +a 3 + +a n = a k The sequence of partial sums is called an infinite series and is denoted a +a 2 +a 3 + = a k 2. Infinite Series If there is a real number S such that
2 lim S n = S, n then we say that the series converges to S and we write a n = S. If the sequence {S n } diverges, we say the series sum. a n diverges and has no Note that the sum of a convergent series is not obtained by ordinary addition, but rather as the limit of the sequence of partial sums. The series associate with Zeno is + 2 + 4 +, 3. Example The partial sums of the series are given by the formula S n = 2 k. k=0 Show that S n = 2 2 n Hence the sum of this series is lim S n = lim 2 n n 2 = 2 n The harmonic series is defined as k. 4. Example 2 The partial sums of the harmonic series are given by the formula
3 S n = k. Show geometrically that S n = k > ln(n+). What is lim n S n? 5. Example 2 revisited A second way to see that the harmonic series diverges is to note that 3 + 4 > 4 + 4 = 2 4 = 2 5 + + 8 > 8 + + 8 = 4 8 = 2 9 + + 6 > 6 + + 6 = 8 6 = 2 Thus k > + 2 + 2 + = 6. Linearity Property Ordinary sums have the following important properties (a k +b k ) = a k + and (ca k ) = c a k These properties naturally extend to infinite series. If a k and b k are convergent series, then b k
4 (a k +b k ) = a k + and (ca k ) = c a k b k 7. Rational or Irrational A rational number is a fraction m n where m and n 0 are integers. All rational numbers have periodic decimal expansions. For example, 3 = 0.33333 Irrational numbers are real numbers which cannot be expressed as a fraction. Examples of irrationals: 2, π, e Since the sum and difference of two fractions is again a fraction, we know that If a and b are rational, then so is a+b and a b. What about 7+π? Rational or irrational? 8. Rational + Irrational The answer is that if a is rational and b is irrational, then the sum a + b must be irrational. The reason this is true is that if a+b is rational, then so is (a+b) a = b, which we know is false. What about the sum of two irrationals?
5 9. Convergent Series + Convergent Series The same reasoning applies to infinite series. If a k and b k are convergent, it follows that (a k +b k ) is also convergent, with sum a k + b k. If the series 0. Convergent Series + Divergent Series a k converges and the series (a k +b k ) must diverge. Otherwise the series [(a k +b k ) a k ] = As a consequence we know b k diverges, then b k converges. k + 2 k diverges. Divergent + Divergent Series What if the series a k and b k both diverge? Does (a k +b k ) converge or diverge. The answer is: You can t tell.
6 If a k = k and b k = k, then k + k diverges If a k = k and b k = k, then k + k = 0 converges (to 0). 2. Summary Convergent + Convergent = Convergent Convergent + Divergent = Divergent Divergent + Divergent = Inconclusive 3. The 999 Question Does.999 =? Reasons to say No : They don t look alike. Any number that starts with.9 cannot equal a number that starts with.0. They are close but not equal. Does.999 =? Reasons to say Yes : 4. The 999 Question. Multiply both sides of the equation 3 = 0.3333 by 3. 2. Let x =.9999. Then
7 0x = 9.9999 x = 0.9999 Subtracting gives us: 9x = 9 or x =. 3. Let x =.9999 and y =. If x y, then the midpoint lies between them. What s the midpoint? 5. The 999 Question and series The number.999 can be thought of as the following series: S = 9 0 + 9 00 + 9 000 + This is an example of a geometric series whose first term is a = 9 0 and where the ratio between two successive terms is r = 0 6. Summing a geometric series In general a geometric series looks like S = a+ar+ar 2 +ar 3 + The partial sum of a geometric series is S n = ar k k=0 Note we start the series with k = 0 7. Summing a geometric series To find a formula for S n, multiply by r:
8 rs n =ar+ar 2 +ar 3 + +ar n +ar n+ S n = a+ar+ar 2 +ar 3 + +ar n Subtacting the second equation from the first. Every term cancels except ar n+ and a. Thus, rs n S n = ar n+ a = (r )S n = ar n+ a = S n = arn+ a r = S n = a arn+ r 8. Summing a geometric series 2 The partial sum of a geometric series is S n = a arn+ r The series covergences if and only if lim S a ar n+ n = lim n n r The limit exists. lim n rn+ = 0 if and only if < r < in which case, ar n = a r. n=0 9. Summary Given a geometric series ar n. n=0 If r <, then
9 n=0 ar n = a r Here a is the first term of the series and r is the common ratio. The series S = 9 0 + 9 00 + 9 000 + 20. The 999 Question Settled is a geometric series whose first term is 9 0 and whose common ratio is r = 0 The sum of this series is a r = 9/0 /0 = The series + 2 + 4 +, 2. Zeno s Paradox Settled associated with Zeno is a geometric series whose first term is and whose common ratio is 2 The sum of this series is a r = /2 = 2 22. Example 3: The Cantor Set The Cantor Set is defined as follows: Start with the interval [0,] Remove the middle third (, 2) 3 3 Remove the middle third of each of the two remaining intervals,
0 That is, remove (, 2) and 9 9 (7, 8). 9 9 Four intervals are left. Remove the middle third of each of these. Repeat this process for ever. What is the total length of the removed intervals? 23. Example 4 Find the sum of the series 2 n +3 n 6 n n= 24. Telescoping Series Another important property of finite sums is the telescoping property which states (b k b k+ ) = b b n+ To extend this property to infinite sums we are led to consider series which These series are known as telescoping series. They converge if and only if the sequence {b k } converges, in which case (b k b k+ ) = b lim n b n for Find the sum 25. Example 5.
n= n 2 +n Trick: Use the partial fraction decomposition of a n = n 2 +n = a n + b n+ Does the sum ( n ) ln n+ n= converge? Trick: Use the property of logs: ( n ) ln = lnn ln(n+) n+ 26. Example 6.