<s>[INST] Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$ Construct a convergent series of positive terms with $\displaystyle\limsup_{n\to\infty} {a_{n+1}\over{a_n}}=\infty$

My thoughts:
By the theorem: Suppose $a_n\ge0$ for all $n$, and let $l=\displaystyle\limsup{\sqrt[n]{a_n}}$. If $l<1$, then $\displaystyle\sum_{n=1}^\infty a_n$ converges; and if $l>1$, then $\displaystyle\sum_{n=1}^\infty a_n$ diverges.
Thus, we know $\displaystyle\sum_{n=1}^\infty {a_{n+1}\over{a_n}}$ diverges
I guess it might works, $(x_n)_{n=1}^{+\infty}$, $x_{n+1} = x_n^2 + x_n$ for all $n\ge1$..
 [/INST] Try the sequence $( \frac{1}{2})^3 + ( \frac{1}{2})^2+ ...+ ( \frac{1}{k})^3+ (\frac{1}{k})^2+ ...$.
Since $\frac{a_{2n+2}}{a_{2n+1}}=\frac{(\frac{1}{n+1})^2}{(\frac{1}{n+1})^3} = n+1$, we have $\limsup_n \frac{a_{n+1}}{a_n} = \infty$. The sequence is convergent since both $\sum \frac{1}{k^2}$ and $\sum \frac{1}{k^3}$ are convergent.
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