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# How to find Radius of Convergence for Sum of ((x-3)^n)/(n3^n) from n =1 to inf?

Please explain and show math. Very Confused.

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- kbLv 78 years agoFavorite Answer
Use the Ratio Test.

r = lim(n→∞) |[(x-3)^(n+1)/((n+1) 3^(n+1))] / [(x-3)^n/(n * 3^n)]|

..= (1/3) |x - 3| * lim(n→∞) n/(n+1)

..= (1/3) |x - 3|.

So, the series converges for r = (1/3)|x - 3| < 1 ==> |x - 3| < 3, and diverges for |x - 3| > 3.

Hence, the radius of convergence is 3.

I hope this helps!

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