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Determine the radius of convergence of ∑ (1 + 1/n)n2 zn

Determine the radius of convergence r of the power series:

    \[ \sum_{n=1}^{\infty} \left( 1 + \frac{1}{n} \right)^{n^2} z^n. \]

Test for convergence at the boundary points if r is finite.


We have

    \[ a_n = \left( 1 + \frac{1}{n} \right)^{n^2} z^n \quad \implies \quad a_n^{\frac{1}{n}} = \left( 1 + \frac{1}{n} \right)^n z. \]

Therefore,

    \begin{align*}  \lim_{n \to \infty} a_n^{\frac{1}{n}} &= \lim_{n \to \infty} \left( 1 + \frac{1}{n} \right)^n z \\[9pt]  &= e\cdot z. \end{align*}

Hence, r = \frac{1}{e} and the series converges for |z| < \frac{1}{e}.

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