Thread: Finding the sum of a series View Single Post
 Sci Advisor HW Helper P: 1,593 Finding the sum of a series It's a very interesting question. There are various techniques for solving such. For example, $$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$ This can be solved by using Fourier coefficients and Parseval's Theorem. $$\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}k=ln(2)$$ is shown by considering: $$f(x)=ln(1+x)$$ and differentiating and considering the Taylor series (thanks Daniel). $$\sum_{k=1}^{\infty}\frac{1}{4k(2k-1)}$$ is solved by considering the sum: $$S=\frac{1}{2}[(1-1/2)+(1/3-1/4)+(1/5-1/6)+...$$ and: $$\sum_{n=0}^{\infty} ne^{-an}$$ is solved by considering: $$z=\sum_{n=0}^{\infty}w^n$$ with: $$w=e^{-a}$$ and differentiating both the sum and the expression for the sum of the corresponding geometric series with respect to w. Tons more I bet. Would be nice to have a compilation of the various methods for calculating infinite sums.