<s>[INST] Series 1 - 1/2^2 + 1/3 - 1/4^2 + 1/5 - 1/6^2 + 1/7... This is the exercise 2.7.2 e) of the book "Understanding Analysis 2nd edition" from Stephen Abbott, and asks to decide wether this series converges or diverges:
$$ 1-\dfrac{1}{2^2}+\dfrac{1}{3}-\dfrac{1}{4^2}+\dfrac{1}{5}-\dfrac{1}{6^2}+\dfrac{1}{7}-\dfrac{1}{8^2}\dots $$
I have noticed that
$$\dfrac{1}{3}< 1-\dfrac{1}{2^2}+\dfrac{1}{3};\\
\dfrac{1}{3}+\dfrac{1}{5}<1-\dfrac{1}{2^2}+\dfrac{1}{3}-\dfrac{1}{4^2}+\dfrac{1}{5};\\
\dfrac{1}{3}+\dfrac{1}{5}+\dfrac{1}{7}<1-\dfrac{1}{2^2}+\dfrac{1}{3}-\dfrac{1}{4^2}+\dfrac{1}{5}-\dfrac{1}{6^2}+\dfrac{1}{7}\\
$$
which is true in general because
$$1-\dfrac{1}{4}-\dfrac{1}{16}-\dfrac{1}{36}-\dfrac{1}{64}-\dots=1-\sum_{n=1}^\infty \dfrac{1}{(2n)^2}=1-\dfrac{1}{4}\sum_{n=1}^\infty\dfrac{1}{n^2}=1-\dfrac{\pi^2}{24}>0.
$$
Thus
$$
\sum_{n=1}^\infty \dfrac{1}{2n+1}<1-\dfrac{1}{2^2}+\dfrac{1}{3}-\dfrac{1}{4^2}+\dfrac{1}{5}-\dfrac{1}{6^2}+\dfrac{1}{7}-\dfrac{1}{8^2}\dots
$$
Finally, as the series $\sum_{n=1}^\infty \dfrac{1}{2n+1}$ diverges, so does the series requested.
My two questions are:
I) Is this reasoning correct?
II) Can this exercise be done without using that $\sum_{n=1}^\infty\dfrac{1}{n^2}=\dfrac{\pi^2}{6}$? I would like to find a solution with more elementary tools.
Thank you.
 [/INST] The basic idea is good. But after you've proven that $\sum_{n=1}^N\frac1{2n+1}$ is smaller that the sum of the first $2N+1$ terms of your series, you're done. And all you need for that is that$$\frac1{2^2}+\frac1{4^2}+\cdots+\frac1{(2N)^2}<1.$$This is equivalent to$$\frac1{1^2}+\frac1{2^2}+\cdots+\frac1{N^2}<4,$$which in turn follows from\begin{align}\frac1{1^2}+\frac1{2^2}+\cdots+\frac1{N^2}&<1+\frac1{1\times2}+\frac1{2\times3}+\cdots+\frac1{(N-1)N}\\&=1+1-\frac12+\frac12-\frac13+\cdots+\frac1{N-1}+\frac1N\\&=2+\frac1N\\&<4.\end{align}
</s>