What is the interval of convergence of $sum_1^oo n!(2x-1)^n$ ?

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Answer 1 · 2016-08-06T22:17:22

Converges only for $x = 1/2$

$sum_1^oo n!(2x-1)^n equiv sum_1^oo n!y^n$ with $y = 2x-1$

With $y = y_0 ne 0$ we have

$lim_{n->oo}n!abs(y_0)^n= oo$ because $EE n_0 < n and n_0 > abs(y_0)$

Concluding, the convergence radius comprehends only the point $y = 0$ or $2x-1=0->x = 1/2$

What is the interval of convergence of sum_1^oo n!(2x-1)^n sum_1^oo n!(2x-1)^n ?