Let $\{a_n\}$ be an increasing sequence of positive real numbers such that the series $\sum\limits_{k=1}^\infty a_k$ is divergent.
Let $s_n=\sum\limits_{k=1}^n a_k$ for $n=1,2,\dotsc$ and $t_k=\sum\limits_{k=2}^n\dfrac{a_k}{s_{k-1}s_k}$ for $n=2,3,\dotsc$ Then $\lim\limits_{n\to\infty} t_n$ is equal to
(a) $\dfrac{1}{a_1}$
(b) $0$
(c) $\dfrac{1}{a_1+a_2}$
(d) $a_1+a_2$
I am totally lost on how to solve this question, please help me!
On
Clark's hint above will put you on the correct track to actually solve this.
But since this is multiple guess....we don't need to be rigorous, we just need to eliminate what is clearly wrong.
First of all, it since every $a_k > 0$, it should be obvious that $\frac {a_k}{s_k s_{k-1}}>0$ and (b) is right out.
Now, how about you pick a sequence that is really easy to work with.. how about $\{1,1,1,1,1,1,1,1,\cdots\}$
Then the first few terms of the series are $\frac 12 + \frac 16 + \frac 1{12} \dots$
And the partial sums are $\frac 12, \frac 23, \frac 34, \frac 45\cdots$
And that leaves only one plausible answer.
hint
Notice that $a_k = s_k - s_{k-1}$