Section 10.2: Series. Definition: An infinite series is a sum of the terms of an infinite sequence {a n } n=1.

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1 Section 02: Series Definition: An infinite series is a sum of the terms of an infinite sequence {a n } a n = a + a 2 + a a n + The nth partial sum of the series is given by s n = a + a 2 + a a n = That is, s = a s 2 = a + a 2 s 3 = a + a 2 + a 3 n a j These partial sums form a new sequence {s n }, called the sequence of partial sums j= Definition: The series a n is said to converge with sum S if the sequence of partial sums {s n } converges to S Otherwise, the series diverges Example: List the first four partial sums of n Does the series converge or diverge? The partial sums of this series are s = s 2 = + = 2 s 3 = + + = 3 s 4 = = 4 s n = n The sequence of partial sums diverges: Therefore, the series diverges lim s n = lim n =

2 Example: List the first four partial sums of The partial sums for this series are In general, s = ) n Does the series converge or diverge? s 2 = + = 0 s 3 = + = s 4 = 0 s n = {, n odd 0, n even Since the limit lim s n does not exist, the series diverges Example: Show that the series 2 n converges The partial sums for this series are ) s = 2 s 2 = = 3 4 s = 2 s 2 = 2 ) 2 s 3 = 2 3 ) s 3 = = 8 s n = s 2 n n = ) 2 n The sequence of partial sums is convergent lim s n = lim 2 ) n Thus, the series converges with sum = Example: If the nth partial sum of the series and a general formula for the nth term a n a n is s n = n +, find the sum of the series 2n + 4 The sum of the series is lim s n = lim n + 2n + 4 = 2

3 The general term a n is given by a n = s n s n = n + 2n + 4 n 2n + 2 = n + 2n + 2) n 2n + ) = n + )n + ) nn + 2) = 2n + )n + 2) 2n + )n + 2) Note: There are certain types of series whose sum can be computed easily, provided that the series is convergent Definition: A series is called a telescoping series if there is an internal cancellation in the partial sums Example: Determine whether the given series converge If so, find the sum of the series a) cos ) n cos n + The nth partial sum of this series is s n = cos cos ) + cos 2 2 cos ) + 3 Since = cos cos n + ) lim s n = lim cos cos = cos, n + the series converges with sum cos cos 3 cos ) + + cos ) 4 n cos n + b) ) n + ln n + 2 Using properties of logarithms, ) n + ln = n + 2 [lnn + ) lnn + 2)]

4 The nth partial sum of this series is s n = ln 2 ln 3) + ln 3 ln 4) + ln 4 ln ) + + [lnn + ) lnn + 2)] = ln 2 lnn + 2) ) 2 = ln n + 2 Since lim s n =, the series diverges c) nn + ) Using partial fraction decomposition, nn + ) = n ) n + The nth partial sum of this series is s n = ) ) ) ) + + n ) = n + n + Since lim s n =, the series converges with sum Definition: A geometric series is a series of the form ar n = a + ar + ar 2 + ar 3 +, where a is a constant and r is called the ratio of the series Theorem: Geometric Series Test) If r <, the geometric series arn, where a 0, converges with sum If r, the series diverges a r Proof: The nth partial sum of the series is s n = a + ar + ar ar n

5 Then rs n = ar + ar 2 + ar ar n and If r <, then rs n s n = ar + ar 2 + ar ar n a + ar + ar ar n ) s n r ) = ar n ) s n = arn ) r lim s n = a r If r >, then lim s n = If r =, then s n = an and {s n } diverges If r =, then {s n } = {a, 0, a, 0, } and lim s n does not exist Example: Determine whether the given series converge If so, find the sum a) 2 ) n 3 Since r = 3 <, the series converges with sum 2 3 = 2 ) 3 = b) ) n 2 The series can be rewritten as ) n+ 2 = 2 ) ) n 2 = 0 ) n 2 Since r = 2 <, the series converges with sum 0 2 = 0 ) = 2

6 c) 2 3n n+ The series can be rewritten as 2 3 ) n = n ) n 8 Since r = 8 >, the series diverges d) The series can be rewritten as ) + = n 2 2) Since r = 2 <, the series converges with sum = 2 ) 2 = 3 3 Example: Find all values of x for which the series the series for these values of x x 3) n converges Find the sum of By the Geometric Series Test, the series converges if x 3 < That is, < x 3 < 2 < x < 4 For these values of x the sum is x 3 x 3) = x 3 4 x

7 Note: In the following sections, we will discuss more general series and convergence tests convergence The following theorem provides a quick way to determine if a series diverges Theorem: The Divergence Test) The series a n diverges if lim a n 0 Proof: Suppose the series a n converges with sum S Then It follows that lim n = S, lim n = S lim a n = lim s n s n ) = S S = 0 We have shown that if a n converges, then lim a n = 0 Thus, if lim a n 0, the series a n diverges Note: If lim a n = 0, the Divergence Test does not provide any information Example: Determine whether the following series converge or diverge a) ) 2n + 3 ln n The series diverges by the Divergence Test ) ) 2n lim ln = ln 0 n b) tan n) The series diverges by the Divergence Test lim tan n) = π 2 0

8 c) n ln n The series diverges by the Divergence Test d) + ) n n lim n ln n = lim n = lim n = The series diverges by the Divergence Test lim + n) n = e 0

To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes.

To discuss this topic fully, let us define some terms used in this and the following sets of supplemental notes. INFINITE SERIES SERIES AND PARTIAL SUMS What if we wanted to sum up the terms of this sequence, how many terms would I have to use? 1, 2, 3,... 10,...? Well, we could start creating sums of a finite number