Complex Variables and Partial Differential Equations (3130005)

Question-2c(ii)

BE | Semester-3 Summer-2020 | 27-10-2020

Q2) (c(ii))

Find the radius of convergence of $\sum_{1}^{\infty} {(1 + \frac{1}{n})}^{n^{2}} z^{n}$ .

Here,

a_{n} = {(1 + \frac{1}{n})}^{n^{2}}

\Rightarrow {(a_{n})}^{\frac{1}{n}} = {(1 + \frac{1}{n})}^{n}

Now,

R = \lim_{n \to \infty} \frac{1}{{(a_{n})}^{\frac{1}{n}}}

R = \frac{1}{\lim_{n \to \infty} {(a_{n})}^{\frac{1}{n}}}

R = \frac{1}{\lim_{n \to \infty} {(1 + \frac{1}{n})}^{n}}

R = \frac{1}{e} [∵ \lim_{n \to \infty} {(1 + \frac{1}{n})}^{n} = e]

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