Prove: If $\sum a_n$ converges, then $\sum \frac{1}{a_n}$ diverges.
I want to prove this statement. I've been trying to find a way but I couldn't.
let's say $a_n$ converges to $a$ then what should i do? Can i prove it like a sequence. For example $\forall\epsilon >0, \exists N>0 \ if \ n>N \ then \ |a-L|< \epsilon$ . I don't think i can apply this to series.
Best Answer
If $\sum a_n$ converges, we must have that $\lim\limits_{n\to\infty}a_n=0$ (otherwise it would have been divergent when we checked the limit by the divergence/limit test). Therefore, if we considered $\sum\frac{1}{a_n}$, when we take the limit, we see that $$\lim_{n\to\infty}\frac{1}{a_n}\left[\to\frac{1}{0}\right]\to\infty$$