$(a_n) $ is a sequence of positive real numbers. The series $\sum a_n$ will converge if
(a) $\sum a_n^2$ converges.
(b)$\sum \frac{a_n}{2^n}$ converges
(c)$\sum \frac{a_{n+1}}{a_n}$ coverges
(d)$\sum \frac{a_n}{a_{n+1}}$ converges
a) can't be true, counter example : $\sum\frac{1}{n^2}$ converges but not $\sum \frac1n$
b) can't be true, counter example : $\frac{n}{2^n}$ converges but not $\sum n$
I can't decide between c and d. I think c might be true by taking $a_n = \frac{1}{(2n)!}$
also I think taking $a_n = (2n)!$ will disprove d also. So is c the correct option?
If $\sum \frac {a_{n+1}} {a_n}$ converges then $\frac {a_{n+1}} {a_n} \to 0$ so $\sum a_n$ converges by ratio test. If $\sum \frac {a_n} {a_{n+1}}$ converges then $\frac {a_n} {a_{n+1}} \to 0$ and $\frac {a_{n+1}} {a_n} \to \infty $, so ratio test tells you that $\sum a_n$ diverges.