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Question Number 6267 by sanusihammed last updated on 21/Jun/16
$${Verify}\:{the}\:{convergence}\:{of}\:{the}\:{exponential}\:{series}\:\:{e}^{{x}} \\ $$$${using}\:{D}'{Alermbert}\:{ratio}\:{test}. \\ $$
Commented by Yozzii last updated on 21/Jun/16
$${u}_{{r}} =\frac{{x}^{{r}} }{{r}!}\Rightarrow\frac{{u}_{{r}+\mathrm{1}} }{{u}_{{r}} }=\frac{{x}^{{r}+\mathrm{1}} {r}!}{\left({r}+\mathrm{1}\right)!{x}^{{r}} }=\frac{{x}}{{r}+\mathrm{1}} \\ $$$$\therefore\:\underset{{r}\rightarrow\infty} {\mathrm{lim}}\mid\frac{{u}_{{r}+\mathrm{1}} }{{u}_{{r}} }\mid=\underset{{r}\rightarrow\infty} {\mathrm{lim}}\frac{\mid{x}\mid}{{r}+\mathrm{1}}=\frac{\mid{x}\mid}{\infty}=\mathrm{0}<\mathrm{1} \\ $$$$\because\:\underset{{r}\rightarrow\infty} {\mathrm{lim}}\mid\frac{{u}_{{r}+\mathrm{1}} }{{u}_{{r}} }\mid<\mathrm{1}\:{for}\:{any}\:{x}\in\mathbb{C},\:{then}\:{e}^{{x}} =\underset{{r}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{x}^{{r}} }{{r}!}\: \\ $$$${is}\:{convergent}\:{for}\:{all}\:{x}\:{according}\:{to} \\ $$$${D}'{Alambert}'{s}\:{Ratio}\:{test}. \\ $$
Commented by sanusihammed last updated on 21/Jun/16
$${Thanks}\:{for}\:{help} \\ $$