Question

(a) Show that the given alternating series converges.

(b) Compute S10 and use Theorem 7.38 to find an interval

containing the sum L of the series.

(c) Find the smallest value of n such that Theorem 7.38

guarantees that Sn is within 10^-6 of $L$

localid="1654683796492" $\sum _{\mathrm{k}=2}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}\mathrm{k}!}{(2\mathrm{k}+1)!}$

Short answer

Part(a) The Given series is

Part (b)

Step-by-step solution

Part(a)  Given information

The series $\sum _{\mathrm{k}=2}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}\mathrm{k}!}{(2\mathrm{k}+1)!}$is given.

We have to show that the given alternating series converges.

Showing the series is Converging

The series $\sum _{\mathrm{k}=2}^{\infty }\frac{{(-1)}^{\mathrm{k}+1}\mathrm{k}!}{(2\mathrm{k}+1)!}$is an alternating series.

Since e