# A limit of a power series

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.

• The first limit should be clear, e.g,. because almost all coefficients of $ef(ex)$ are $>\frac 12$. – Hagen von Eitzen Jun 5 at 5:59
• I agree. Many thanks to Eitzen. – seoneo Jun 5 at 8:09
• 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. – LutzL Jun 5 at 13:08