For what $x's$ does $\sum_{k=1}^{\infty} \frac{(k+1)^{k^2}}{k^{k^2+2}} x^k$ converge?

real-analysis sequences-and-series calculus

Solution 1:

For the case $x=1/e$, a hint: Write the $k$th term as $$\left(\frac {k+1}k\right)^{k^2}\frac1{e^kk^2},$$ then apply the inequality $1+t\le e^t$ to $\frac{k+1}k$.

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