11.2 Convergent or divergent series Rita Korsunsky

Series vs. Sequence: Sequence is a collection of numbers that are in one to one correspondence with the positive integers. 0.6 , 0.06 , 0.006 , 0.0006 , … Series is an expression that represents an infinite sum of numbers. 0.6 + 0.06 + 0.006 + 0.0006 + 0.00006 + … kth partial sum Sk of the series an Sk = a1 + a2 + … + ak. s1 = 0.6 s2 = 0.6 + 0.06 = 0.66 s3 = 0.6 + 0.06 + 0.006 = 0.666 s4 = 0.6 + 0.06 + 0.006 + 0.0006 = 0.6666 … Sequence of partial sums of an S1, S2, S3, …, Sn, …. 0.6, 0.66, 0.666, 0.6666, 0.66666, …. Series an converges if sequence of partial sums {Sn} converges an diverges if {Sn} diverges. A divergent series has no sum. 0.6, 0.66, 0.666, 0.6666, 0.66666, ….  2/3 0.6 + 0.06 + 0.006 + 0.0006+…

Example 1 series converges, sum= 1. series diverges. Given the series: (Telescoping series) Find S1, S2, S3, S4, S5, and S6. Find Sn. Converges or diverges? Solution: (a) S1 = 1, S2 = 1 + (-1) = 0, S3 = 1 + (-1) + 1 = 1, S4 = 0, S5 = 1, S6 = 0. (b) (c) (c) series converges, sum= 1. series diverges.

Prove: The Harmonic Series is a divergent series. ( ) ( ) ( ) ( ) Group: S4 S8 S16 Notice that: {Sn} diverges, so series is divergent.

Proof:

Example 3 Prove that the following series converges, and find its sum: SOLUTION: Geometric, r = 0.1. |r| < 1, so it converges to SOLUTION: Geometric, |r| = 1/3 < 1, so it converges to

Example 4: Find all values of x for which converges, and find the sum of the series.

If a series an is convergent, then Theorem: If a series an is convergent, then PROOF: n th term test: Illustration: Series nth term test Conclusion Diverges by nth term test Further investigation necessary Further investigation necessary Diverges by nth term test

Theorem Example 4 For any positive integer k, the series either both converge or both diverge. Example 4 Show that the following series converges: Solution: Delete the first two terms of the telescoping series which converges. The series converges.

If an and bn are convergent series with sums A and B, respectively, Theorem If an and bn are convergent series with sums A and B, respectively, Example 5 Prove that the series converges, and find its sum: Series converges and has the sum 7 + 3 = 10

Diverges Theorem Example 6 If an is a convergent series and bn is divergent, then (an + bn) is divergent Example 6 Determine if diverges or converges. Geometric, |r| = 1/5 < 1, converges Divergent harmonic Diverges

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