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

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Answer 1 · 2016-01-28T23:14:30

#(1/2,3/2]#

Use the ratio test.

The infinite sum #sum^ooa_n# converges when

#lim_(nrarroo)abs((a_(n+1))/(a_n))<1#

This gives

#lim_(nrarroo)abs(((-2)^(n+1)(n+2)(x-1)^(n+1))/((-2)^n(n+1)(x-1)^n))<1#

Simplified:

#lim_(nrarroo)abs((-2(n+1)(x-1))/(n+2))<1#

Evaluating the limit yields

#abs(-2(x-1))<1#

Resulting in the inequality

#1/2 < x < 3/2#

We now have to plug in #x=1/2# and #x=3/2# to see if the sum converges at the endpoints.

Plugging in #x=1/2# does not converge, but #x=3/2# does.

So, the answer is

#1/2 < x <=3/2# or #(1/2,3/2]#