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how can you simplify (n^10)/((n+1)^10)?

This is coming from the original question of :

Find all the values of x such that the given series would converge.

summation of ((11x)^n)/(n^10) from n=1 to infinity

The series is convergent
from ?, left end included (enter Y or N):?
to ?, right end included (enter Y or N):?

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1 Answer

Convergent means

lim{n->oo} |[(11x)^(n+1))/(n+1)^10)]/[(11x)^n)/(n^10)]| < 1

|11x| < 1, attn: lim{n->oo}(n^10)/((n+1)^10) = lim{n->oo} 1/((1+1/n)^10) = 1

R = 1/11

At x = +/-1/11

summation of |((11x)^n)/(n^10)| from n=1 to infinity converges by p-series test.

Answer: The series is convergent on [-1/11, 1/11].