Is there a better way to do this problem?

For the following convergent series, at least how many terms do you have to sum so that the partial sum is within 0.001 of the sum of the series?

If you add the first k terms, the remainder is

Using the fact that

take the derivative of both sides

let x=1

so we want

but that can't be solved explicitly for k

although it appears to be true for k >8