What is the interval of convergence of #sum_1^oo ((-1)^(n-1)*x^(n+1))/((n+1)!)#?

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Answer 1 · 2018-04-30T18:05:32

Use the ratio test to detmeurine the radius of convergence first:

#L = lim_(n-> oo) (((-1)^(n)x^(n + 2))/((n + 2)!))/(((-1)^(n - 1)x^(n + 1))/((n + 1)!)#

#L = lim_(n->oo) (-1(x))/(n + 2)#

Take the absolute value.

#L = |x|lim_(n->oo) 1/(n + 2)#

#L = |x| (0)#

#L = 0#

This converges for all values of #x#, because the ratio test came back between #0# and #1#.

The interval of convergence is therefore #(-oo, oo)#.

Hopefully this helps!