Your serie is equal to
Consider So,
Finally
Now set and we're done.
hi, i have a series 1-1/3+1/5-1/7+..., which i think is also called the Leibniz formula, i know that to find the nth term you can can use (-1)^n/(2n+1), but i need help trying to find a formula or a way to find the sum of the series for a certain number of terms i.e. the sum of the first 10 terms, can anyone help?
Proof 1: First note that for . This tells us that (integrate both sides) for . Now to justify what Krizalid did is to use Abel's Theorem which tells us that the series is continous at because it converges there (by alternating series test). Thus, if is a sequence in then thus .
Abel's Theorem: If is a power series of radius of convergence of radius such that the series converges at then is continous at .
Here is a more elementary proof.
Proof 2: For any we have . Thus, where depends on the parity of . This tells us that . Thus, . Now . This means, . This means for any if we chose it will mean that is within of . And so, by definition, this series converges to .