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Suggestion of how to do it, please.

Suppose $\{a_n\}$ is a succession in $\mathbb R ^+$ such that $\sum a_n$ diverges, and if $s_n = \sum\limits_{k=1}^n{a_k}$. Show that $\sum\limits_{n=1}^\infty{ \frac {a_n}{1+a_n}}$ diverge.

Please.thank

VERA
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  • what's the point in the definition of $s_n$ – mathworker21 Oct 12 '18 at 04:58
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    if $\sum_n \frac{a_n}{1+a_n}$ converges, then $a_n \to 0$ which implies $\frac{a_n}{1+a_n} \ge \frac{1}{2}a_n$ for $n$ large, so compare to $\sum a_n$ which diverges – mathworker21 Oct 12 '18 at 04:59
  • $s_n$ is is the limit of the succession of sums – VERA Oct 12 '18 at 05:01
  • I specifically asked you what the point of defining it is. You didn't give an answer, presumably because there is no point to defining it. Also, my second problem gives a proof of your problem statement – mathworker21 Oct 12 '18 at 05:03
  • You can see https://math.stackexchange.com/questions/131678/positive-series-problem-sum-limits-n-geq1a-n-infty-implies-sum-n-geq1 – Riemann Oct 12 '18 at 05:36

2 Answers2

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Suppose $a_n\to0$; otherwise $\frac{a_n}{1+a_n}\not\to0$. Then since $$\frac{a_n}{\frac{a_n}{1+a_n}}\to1,$$by ratio test, these two series share the convergence result.

JRen
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Theorem Suppose that we have two series $ \Sigma_n a_n $ and $\Sigma_n b_n$ with $ a_n\geq 0, b_n > 0 $ for all $ n$. Then if $ \lim_{n \to \infty} \frac{a_n}{b_n} = c$ with $ 0 < c < \infty $ then either both series converge or both series diverge..

$\displaystyle \frac{a_n}{b_n} = \frac{a_n}{\frac{a_n}{a_n+1}} = a_n+1 \to 1 $ as $a_n \to 0$. If $a_n \not \to 0$ then the second series clearly diverges.

Migos
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