Verify the convergence of the exponential series e^x using D′Alermbert ratio test.


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Question Number 6267 by sanusihammed last updated on 21/Jun/16

$V e r i f y t h e c o n v e r g e n c e o f t h e e x p o n e n t i a l s e r i e s e^{x}$ $u s i n g D^{'} A l e r m b e r t r a t i o t e s t .$
Commented by Yozzii last updated on 21/Jun/16

$u_{r} = \frac{x^{r}}{r!} \Rightarrow \frac{u_{r + 1}}{u_{r}} = \frac{x^{r + 1} r!}{(r + 1)! x^{r}} = \frac{x}{r + 1}$ $∴ \lim_{r \to \infty} ∣ \frac{u_{r + 1}}{u_{r}} ∣= \lim_{r \to \infty} \frac{∣ x ∣}{r + 1} = \frac{∣ x ∣}{\infty} = 0 < 1$ $∵ \lim_{r \to \infty} ∣ \frac{u_{r + 1}}{u_{r}} ∣< 1 f o r a n y x \in C, t h e n e^{x} = \underset{r = 0}{\sum^{\infty}} \frac{x^{r}}{r!}$ $i s c o n v e r g e n t f o r a l l x a c c o r d i n g t o$ $D^{'} {A l a m b e r t}^{'} s R a t i o t e s t .$
Commented by sanusihammed last updated on 21/Jun/16

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