Question

If the series is convergent, use the Alternating Series Estimation Theorem to determine how many terms we need to add in order to find the sum with an error less than 0.000005. (If the quantity diverges, enter DIVERGES.) 6 Incorrect: Your answer is incorrect. terms Need Help? Read It Watch It

Answer 1

Answer:

Number of terms required is 5 terms

Step-by-step explanation:

Given the series: $\sum\imits^{\infty}_{n=0}\frac{(-1)^n}{5^nn!}$

Given the series is an alternating series,

                                   $b_n=\frac{1}{5^nn!}$

Evaluating the limit: $\lim_{n \to \infty} b_n= \lim_{n \to \infty} (\frac{1}{5^nn!})=\frac{1}{\infty}=0$

Since $\lim_{n \to \infty} b_n =0 \,\, and \,\, b_{n+1}\leq b_n$ for all n = the series is convergent

The error of an alternating series $\sum b_n$ is bounded as

$|R_n|\leq b_{n+1}$

Given $b_n = \frac{1}{5^n n!}=b_{n+1}=\frac{1}{5^{n+1}(n+1)!}200000$

By trial and error: the above equation is satisfied for $n=4$

Since the given series starts at n = 0, the number of terms required is 5 terms