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How to solve the following problem?

Let {${a_n}$} be an increasing sequence of positive real numbers such that the series $\sum_{k=1}^{\infty}{a_k}$ is divergent. Let $S_n$=$\sum_{k=1}^{n}{a_k}$ for $ n=1,2,...$ and $t_n$=$\sum_{k=2}^{n}{\frac{a_k}{S_{k-1}S_k}}$ for $n=2,3,... $

Then $lim_{n\to\infty}{t_n}$ is equal to

(a) $\frac{1}{a_1}$

(b) $0$

(c) $\frac{1}{a_1+a_2}$

(d) $a_1+a_2$

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closed as off-topic by Did, Martin R, user296602, Namaste, Saad Mar 28 '18 at 1:00

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It would be appreciated if the homework questions policy was observed before asking this question.

Others have pointed towards different versions of this question, but following the policy of giving a hint at first and a full answer when the homework is expected to be due (for completeness of this site), here is your hint:

$$ a_k = S_k-S_{k-1} $$

So: $$ \frac{a_k}{S_k S_{k-1}}=\frac{S_k-S_{k-1}}{S_k S_{k-1}}=\frac{1}{S_{k-1}}-\frac{1}{S_k} $$

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  • 1
    $\begingroup$ Exactly that hint was given at math.stackexchange.com/a/1115979. $\endgroup$ – Martin R Mar 27 '18 at 14:00
  • $\begingroup$ @MartinR Yes indeed. $\endgroup$ – Mefitico Mar 27 '18 at 14:27
  • $\begingroup$ oh! that's pretty easy! I couldn't find or think about this hint before.. Thank you @Mefitico $\endgroup$ – user511110 Mar 27 '18 at 16:31

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