sequences and series - What is the value of $frac11+frac13-frac15-frac17+frac19+frac1{11}-dots$?

The series $\sum_{k=1}^{\infty }\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$ converges to $\frac{\pi}{4}$. Here, the sign alternates every term.

The series $\displaystyle\sum_{k=1}^{\infty }{(-1)^{\left(k^{2} + k + 2\right)/2} \over 2k-1}=\frac{1}{1}+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\dots$ also converges. Here, the sign alternates every two terms.

What is the convergence value, explicitly, of the second series?

The first summation is noted above, because it might be a useful information to evaluate the second summation.

Answer

Let $$\begin{align} S_1&=1-\frac15+\frac19-{1\over13}+\dots\\ S_2&=\frac13-\frac17+{1\over11}-{1\over15}+\dots \end{align}$$ so that the sum we seek is $S_1+S_2.$
T0 compute $S_1,$ consider $$f(x) = 1-{x^5\over5}+{x^9\over9}-{x^{13}\over13}+\dots$$ so that $$f'(x)=-x^4+x^8-x^{12}+\dots={-x^4\over1+x^4},\ |x|<1$$
and $$f(x)=\int_0^x{-t^4\over1+t^4}\mathrm{dt}+f(0),\ |x|<1$$
By Abel's limit theorem, $$S_1=\lim_{x\to1-}\int_0^x{-t^4\over1+t^4}\mathrm{dt}+f(0)=1-\int_0^1{t^4\over1+t^4}\mathrm{dt}$$ and we can do a similar calculation for $S_2$ to get $$S_2=\int_0^1{t^2\over1+t^4}\mathrm{dt}$$

The integrals are elementary, but tedious, and I leave them to you. (Frankly, I would do them by typing them into WolframAlpha. You can get the indefinite integrals, and check them by differentiation if you want to.) If you really want to do them by hand, I think it's easiest to consider only the definite integrals, and split the denominator into linear factors using complex numbers, but I don't know if you've learned about complex integrals yet.