Question

consider the following series.
sum _ { n = 1 } ^ { infty } \frac { ( - 1 ) ^ { n + 1 } } { n ^ { 4 } } ( | \text { error } | < 0.00005 )
show that the series is convergent.
since this series is an alternating series, which condition(s) below show that it converges? (select all that apply.)
\begin{array} { l } { square lim _ { n
ightarrow infty } \frac { 1 } { ( n + 1 ) ^ { 4 } } = 0 } \\ { square \frac { 1 } { ( n + 1 ) ^ { 4 } } > \frac { 1 } { n ^ { 4 } } } \\ { square lim _ { n
ightarrow infty } \frac { 1 } { n ^ { 4 } } = 0 } \\ { square \frac { 1 } { ( n + 1 ) ^ { 4 } } < \frac { 1 } { n ^ { 4 } } } end{array}
how many terms of the series do we need to add in order to find the sum to the indicated accuracy?
terms

Explanation:

Step1: Check convergence conditions for alternating series

For an alternating series \(\sum_{n = 1}^{\infty}(-1)^{n+1}a_n\) (\(a_n=\frac{1}{n^4}\)), by the Alternating Series Test:

1. We need \(\lim_{n

ightarrow\infty}a_n = 0\). Since \(a_n=\frac{1}{n^4}\), \(\lim_{n
ightarrow\infty}\frac{1}{n^4}=0\).

1. We need \(a_{n + 1}

Step2: Use the error bound for alternating series

For an alternating series \(\sum_{n = 1}^{\infty}(-1)^{n+1}a_n\), the error \(|R_N|\leq a_{N + 1}\). We want \(|R_N|<0.00005\), so we set \(a_{N+1}=\frac{1}{(N + 1)^4}<0.00005\).
Solve the inequality \(\frac{1}{(N + 1)^4}<0.00005\), or \((N + 1)^4>\frac{1}{0.00005}=20000\).
Take the fourth - root of both sides: \(N+1>\sqrt[4]{20000}\approx11.89\). Since \(N\) is an integer, \(N + 1=12\), so \(N = 11\).

Answer:

For the convergence conditions: \(\lim_{n
ightarrow\infty}\frac{1}{n^4}=0\) and \(\frac{1}{(n + 1)^4}<\frac{1}{n^4}\) (i.e., the third and fourth options). The number of terms needed is \(11\).