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Convergence of Power Series

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In Probability,

\sum\limits_{j=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

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xn

=

a0+

a1x+

a2x2+

a3x3+

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Find the reason for convergence (RFC) with respect to $x$

e^x = \sum\limits_{n=0}^{\infty} {x^n \over{n!}}

give $a_n = {x^n \over n!}$

L \equiv \lim_{n \to \infty} |{x^{n+1} \over {(n+1)!}} \times {n! \over x^n}|

= \lim_{n \to \infty} |{x \over {n+1}}| = |x| \lim_{n \to \infty} |{1 \over {n+1}}| = |x| \times 0 = 0 < 1

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