Question about convergence of series...?

r = lim(n→∞) |[(n+1)^10 (x+10)^(n+1) / (4^(n+1) (n+1)^(32/3))] / [n^10 (x+10)^n / (4^n n^(32/3))]|
..= (1/4) |x+10| * lim(n→∞) ((n+1)^10/n^10) * n^(32/3)/(n+1)^(32/3)
..= (1/4) |x+10| * lim(n→∞) n^(2/3)/(n+1)^(2/3)
..= (1/4) |x+10| * 1
..= (1/4) |x+10|.

So, the series converges when r = (1/4)|x+10| < 1 ==> |x+10| < 4, plus possible endpoints.

Checking the endpoints:
x = -6 ==> Σ n^10 / n^(32/3) = Σ 1/n^(2/3), divergent p-series
x = -14 ==> Σ (-1)^n n^10 / n^(32/3) = Σ (-1)^n/n^(2/3), convergent by Alternating Series Test,
since {1/n^(2/3)} is decreasing and converges to 0.

So, the interval of convergence is [-14, -6).

I hope this helps!

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