Convergence problem power series?

Sum of ((x-1)^n)/(n3^n) from 1 to infinity

So first I did the ratio test

Lim((x-1)^n+1/(n+1)3^(n+1)) = Lim(n(x-1)/3(n+1))

and got that the series converges when -2< x < 4 and it diverges when the -2 > x or 4 < x.

Now I'm trying to test whether the series converges when x = -2 or 4.

When I used the ratio test, it failed and I'm not sure where to go from there.

Comments

  • I dont think you need to use the ratio test to test x= -2 and 4. When you plug in x=-2, you get (-1)^n/n and you can try applying the alternating series test. When you plug in x=4 you get 1/n which is the harmonic series and it diverges. I find most of the time using other tests other then the ratio test to determine if my interval converges or diverges usually works better.

    GL.

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