Question

Is the given statement true or false?
Radius of convergence of the function f(x)=e^x is 1.
Justify your answer.​

Answer 1

SOLUTION

TO CHECK

True / False below statement :

Radius of convergence of the function $\displaystyle \sf{f(x) = {e}^{x} }$ is 1

EVALUATION

Here the given function is

$\displaystyle \sf{f(x) = {e}^{x} }$

Now the power series expansion is

$\displaystyle \sf{f(x) = {e}^{x} = 1 + x + \frac{ {x}^{2} }{2!} + \frac{ {x}^{3} }{3!} + .. + \frac{ {x}^{n} }{n!} + .. }$

Now the coefficient in n th term

$\displaystyle \sf{a_{n} = \frac{1}{n!} }$

Thus we get

$\displaystyle \sf{a_{n + 1} = \frac{1}{(n + 1)!} }$

Now

Radius of convergence R is given by

$\displaystyle \sf{R = \lim_{n \to \infty } \bigg| \frac{a_n }{a_{n + 1} } \bigg| }$

$\displaystyle \sf{ \implies \: R = \lim_{n \to \infty } \bigg| \frac{(n + 1)!}{n!} \bigg| }$

$\displaystyle \sf{ \implies \: R = \lim_{n \to \infty } \bigg| \frac{(n + 1) \times n!}{n!} \bigg| }$

$\displaystyle \sf{ \implies \: R = \lim_{n \to \infty } \bigg| (n + 1) \bigg| }$

$\displaystyle \sf{ \implies \: R = \infty }$

So the given statement is FALSE

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