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This was one point question on my midterm, however I didn't get the point and the professor didn't explain why and I am wondering what is wrong, the professor just wrote a big NO on the paper.

Let $a_n$ be a positive sequence, prove that $\sum\frac{a_n}{1+a_n}$ converges if and only if $\sum a_n$ converges.

Now, this question shows a couple of times on maths.stackexchange, and is solved in different ways (and I am not asking for a solution), I am wondering what is wrong with this answer

Since both series are positive, the limit comparison test works

Assume the first sum converges, then $a_n \to 0$, and by the limit comparison test, $\lim \frac{a_n}{\frac{an}{1+a_n}}=1$

Assume the 2nd sum converges, then $\frac{a_n}{1+a_n} \to 0 > \Leftrightarrow a_n \to 0$ and as previously, $\lim > \frac{a_n}{\frac{an}{1+a_n}}=1$

I have no idea what is wrong with this

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    $\begingroup$ Is the part in the blockquote exactly what you answered? If so, you should have said that the limit comparison test works in your answer, not only in your question here. In any case, you should have said that $\lim \dfrac{a_n}{\frac{a_n}{1+a_n}} = 1$ if either of the series converges, and hence, by the limit comparison test, each series converges if and only if the other does. It's more a matter of wrongly writing up the argument than of a mathematical error. $\endgroup$ Jun 5, 2017 at 20:31
  • $\begingroup$ @DanielFischer It is exactly (English version, my native language isn't English) what I wrote. A previous part of the exam was this question: $$.$$ Let $f$ is differentiable at zero and $f(0)=0$ and $f'(0) \ne 0$ and let $a_n$ be a positive sequence that tends to zero, prove that $\sum f(a_n)$ converges iff $\sum a_n$ does? $$.$$ Could he wanted me to let $f(x)=\frac{x}{1+x}$ instead? $\endgroup$
    – Rab
    Jun 5, 2017 at 21:08
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    $\begingroup$ Could be that that was the expected answer, but that doesn't make different ways invalid. I think it's the way you wrote up your argument. Perhaps the professor didn't see what you meant, or they were just strict. I'd have given partial marks, the correct idea is there, but it's not written up well. $\endgroup$ Jun 5, 2017 at 21:21
  • $\begingroup$ Why don't you just ask your professor instead? $\endgroup$ Jun 6, 2017 at 9:14
  • $\begingroup$ According to policy, he can not discuss an exam unless its a grade appeal, of which I get once per course (unless the appeal is considered a reasonable appeal and if so I get an extra one). I wanted to make sure I didn't do a trivial mistake and waste mine on a midterm for nothing. $\endgroup$
    – Rab
    Jun 6, 2017 at 9:44

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The proof you wrote proves that the sequence $b_n=\frac{a_n}{a_n+1}$ converges. But you were asked to consider a sequence of partial sums, that is $\sum{\frac{a_n}{a_n+1}}$ is said to converge only if its partial sums converges.

So you should have been considering$$S_n=\sum^{n}{\frac{a_k}{a_k+1}}$$

The reason your professor just wrote a big NO on your paper is because it is a common mistake to get the convergence of sequences and series confused, and he/she likely emphasized that point in class. But it's confusing and its easy to conflate the two.

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    $\begingroup$ The mention of the limit comparison test clearly indicates that it's the convergence of the series that is considered. $\endgroup$ Jun 5, 2017 at 20:32
  • $\begingroup$ What Daniel said. $\endgroup$
    – Rab
    Jun 5, 2017 at 21:00

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