limit of a sequence involving partial sums where the sequence of partial sums diverges [closed]

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) $a_1+a_2$

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Martin R, Community, Namaste, Saad
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• Nice little exercise, what a pity we cannot answer it because you posted no personal input whatsoever... – Did Mar 27 '18 at 11:49
• – Martin R Mar 27 '18 at 11:53

1 Answer

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

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