Chapter 7: Q. 54 (page 653)
In Exercises 52–57, do each of the following:
(a) Show that the given alternating series converges.
(b) Compute $$S_{10}$$ 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 $$S_{n}$$ is within $$10^{−6}$$ of $$L$$.
\[ \sum_{k=0}^{\infty} \dfrac{(-1)^k}{(2k+1)!} \]
Short Answer
(a). Using the test for convergence for alternating series we can see that the given alternating series converges.
(b). The value of $$S_{10}$$ is $$1 - \dfrac{1}{3!} + \dfrac{1}{5!} -\dfrac{1}{7!} + \dfrac{1}{9!} -\dfrac{1}{11!} + \dfrac{1}{13!} - \dfrac{1}{15!} + \dfrac{1}{17!} - \dfrac{1}{19!} + \dfrac{1}{21!} \approx 0.841470984$$
and the value of $$L$$ is $$L \in (0.84147098480789650665254, 0.8414709848078965066526)$$.