summation, any similar function

Question

How can I calculate the second sigma:

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Note: I was told to use Taylor series of: $e^x, sin(x), cos(x), ln(1+x), arctg(x)$

Edit: the problem is that I don't see any of the sigmas similar to a taylor series of a given function to use

Answer 1

The first one is from the taylor series of $\ln(1+x)$: $$\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n}$$ In your case, $x = \frac{1}{3}$ and one needs to add a negative sign to correct between $(-1)^n$ and $(-1)^{n-1}$, so your second sum is $-\ln(1+\frac{1}{3}) = \ln(\frac{3}{4})$. For the first sum, use the fact that $$\cos(x) = \sum_{n=0}^{\infty} (-1)^n\frac{x^{2n}}{(2n)!}$$