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Convergence of Power Series

In Probability,

j=0anxn=a0+a1x+a2x2+a3x3+\sum\limits_{j=0}^{\infty} a_n x^n = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots

Find the reason for convergence (RFC) with respect to xx

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

give an=xnn!a_n = {x^n \over n!}

Llimnxn+1(n+1)!×n!xnL \equiv \lim_{n \to \infty} |{x^{n+1} \over {(n+1)!}} \times {n! \over x^n}| =limnxn+1=xlimn1n+1=x×0=0<1= \lim_{n \to \infty} |{x \over {n+1}}| = |x| \lim_{n \to \infty} |{1 \over {n+1}}| = |x| \times 0 = 0 < 1