Do i need a calculator to find the approximation with the error 10^-8?

The series i got from f(x) is the series from n=0 to inf of (-1)^n(x)^(5n) / n!

Then i evaluated the series' integral from 0 to 0.1.

I wasnt sure how i could go about finding the value of this integral within 10^-8.

I am using the alternating series estimation error to do this but i find it difficult to do it without checking my calculator.