# 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

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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

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