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A question that I came across on my math course's problem sheet:

Let $a_n$ be a sequence of non-negative terms. Show that $\sum_{n=0}^\infty a_n$ conevrges, if and only if $\sum_{n=0}^\infty \frac{a_n}{1+a_n}$ converges.

As $\frac{a_n}{1+a_n} \leq a_n$ for all $n$, I am able to show that convergence of $\sum_{n=0}^\infty a_n$ implies convergence of $\sum_{n=0}^\infty \frac{a_n}{1+a_n}$ by direct comparison test, but I don't see how the information is sufficient to show that convergence of the latter sum implies that of the former.

Could someone please help with the solution, or at least point me in the right direction? Thank you.

stoic-santiago
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    Assume $a_i<1$ for all $i\geq$ some constant. What can you say about the comparaison test now ? – Klangen Aug 12 '19 at 13:07

1 Answers1

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You can show that since $\frac{a_n}{a_n+1}$ goes to zero, $a_n$ also goes to zero. Hence it is bounded by, say, $A\in \mathbb{R^+}$.

Then you can use the comparison test on (since $a_n\le A$)

$$\frac{a_n}{A+1}\le\frac{a_n}{a_n+1}$$

to show that $\sum_{n=0}^\infty a_n$ converges.

Stijn D'hondt
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