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I was trying to make some undergraduate level analysis problem. The point was interval of convergence of power series. It seems that the students are bored with usual coefficients. So I considered the following 'relatively' new kind of power series. $$ f(x) = \sum_{n=1}^\infty \left( 1-\frac{1}{n}\right)^{n^2} x^n $$ The question I proposed is to find the interval of convergence of $f(x)$. Soon after, I realized using this series I can make rather interesting questions.

Prove that $\lim_{x \rightarrow e^-} f(x) = \infty$ and find the limit $\lim_{x \rightarrow -e^+} f(x)$.

I think this maybe an interesting question for undergraduate students. However I have failed to make a reasonable solution. Please help me to improve this questions.

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  • $\begingroup$ The first limit should be clear, e.g,. because almost all coefficients of $ef(ex)$ are $>\frac 12$. $\endgroup$ – Hagen von Eitzen Jun 5 at 5:59
  • $\begingroup$ I agree. Many thanks to Eitzen. $\endgroup$ – seoneo Jun 5 at 8:09
  • $\begingroup$ With the usual $$ (1-n^{-1})^{n^2}=\exp(-n-\tfrac12-\tfrac13n^{-1}+O(n^{-2})) $$ you get, in a first approximation, the value of the series as $$ \frac{e^{-1/2}}{1-e^{-1}x} + \frac{e^{-1/2}}3\ln(1-e^{-1}x) \xrightarrow{x\to -e^+}e^{-1/2}\left(\frac12+\frac13\ln 2\right). $$ One can now refine this by adding the exact differences to the first $N$ terms of the series. $\endgroup$ – LutzL Jun 5 at 13:08

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