Consider a sequence (x) such that
$\frac{x_{n+1}}{x_n}$
this ratio diverges.
Now for divergent series $10^{n^2}$, the ratio diverges.
My question is, is there any convergent series for which the ratio diverges.
I already find out that the limit of the converging sequence must be zero, for satisfying the above condition.
On
Yes (assuming you meant absolute value, otherwise the claim is not true), if $L=\lim |\frac{x_{n+1}}{x_n}|$ exists and there exists an $N$ such that for $n>N$ we have $x_n>0$, then we can test convergence using $L$. $L>1$ implies divergence, $L<1$ implies convergence and the test is inconclusive for $L=1$. Look if you can proof this and if not I can edit later and include a full proof.
To prove the claim, assume first $L<1$ and then choose an $a$ such that $L<a<1$ and obtain an $N$ so that $|x_{n+1}|<a|x_n|$ for $n\geq N$. Then show $|x_n|<a^{n-N}|x_N|$ for $n>N$. The case when $L>1$ follows by considering the sequence $y_n=\frac{1}{|x_n|}$. It should not be too difficult using these hints.
Consider $\frac 1 2 +\frac 1 3 +\frac 1 {2^{2}} +\frac 1 {3^{2}}+\frac 1 {2^{3}} +\frac 1 {3^{3}}+...$. the series converges but $\frac {a_{n+1}} {a_n} \to \infty$ along a subsequence.