Convergence of Power SeriesIn Probability, ∑j=0∞anxn=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 + \cdotsj=0∑∞anxn=a0+a1x+a2x2+a3x3+⋯ Find the reason for convergence (RFC) with respect to xxx Ratio Test Test the boundaries ex=∑n=0∞xnn!e^x = \sum\limits_{n=0}^{\infty} {x^n \over{n!}}ex=n=0∑∞n!xn give an=xnn!a_n = {x^n \over n!}an=n!xn L≡limn→∞∣xn+1(n+1)!×n!xn∣L \equiv \lim_{n \to \infty} |{x^{n+1} \over {(n+1)!}} \times {n! \over x^n}|L≡n→∞lim∣(n+1)!xn+1×xnn!∣ =limn→∞∣xn+1∣=∣x∣limn→∞∣1n+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=n→∞lim∣n+1x∣=∣x∣n→∞lim∣n+11∣=∣x∣×0=0<1
