How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

Note: Enter the smallest possible integer.

sum n=1 to infinity (-1)^n-1(7/n^4), error

Question

How many terms of the series do we need to add in order to find the sum to the indicated accuracy?

Note: Enter the smallest possible integer.

sum n=1 to infinity (-1)^n-1(7/n^4), error <= .001

(This is an alternating series).

blockcolder · Answer

$$\sum_{n=1}^{\infty}(-1)^{n-1}\frac{7}{n^4} \qquad b_n=\frac{7}{n^4}$$
The error of an alternating series at the nth partial sum is less than the (n+1)th term.
Thus, to ensure that the error is less than 0.001, we should have
$$0.001\leq \frac{7}{(n+1)^4}$$
Solve for n here then round up the value to ensure the desired accuracy. :D