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This is a part of a question to show absolute convergence/divergence of improper integrals.

So my professor simplified the question to this $\frac{2}{\pi} \sum\limits_{n=1}^{\infty} \frac{1}{n+1} $ and then proceeded to say that "it is a well known fact that $\sum\limits_{n=1}^{\infty} \frac{1}{n+1} $ diverges".

My question is that shouldn't it converge ? As I can say that the values of the individual terms in the series $\sum\limits_{n=1}^{\infty} \frac{1}{n+1} $ will finally tend to $0$ as $n$ approaches $\infty$.

I cannot ask this in class as the lectures are pre recorded and the doubt session is quite far away.

Shaurya Goyal
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  • Look up the harmonic series. – Gary Jan 14 '21 at 17:36
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    Specifically, see https://math.stackexchange.com/q/255/215011 – grand_chat Jan 14 '21 at 17:38
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    $a_n\to 0$ doesn't imply that the series $\sum\limits_{n=1}^\infty a_n$ converges. Actually, $\sum\limits_{n=1}^\infty\frac{1}{n}$ is the most standard example of a divergent series. – Mark Jan 14 '21 at 17:38
  • You may have heard somewhere that if $\sum\limits_{n=1}^\infty a_n$ converges then $a_n\to 0$. This is a one way implication... $P\implies Q$. This is not a bi-implication. The converse is not logically equivalent and is outright false in this case, the harmonic series being a stereotypical counterexample. – JMoravitz Jan 14 '21 at 17:53

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