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Use the ratio test to solve

Evaluate `sum_(n=1)^(oo)n^(10)/10^n ` We are asked to use the ratio test:


For the ratio test, we take the limit of the ratio of the (n+1)st term to the nth term of the sequence of partial sums. If this limit is less than 1 in absolute value, the series converges, if greater than 1 it diverges, and there is no conclusion if the limit is 1.


`lim_(n->oo)((n+1)^10/(10^(n+1)))/(n^10/10^n) `


`=lim_(n->oo)((n+1)^10)/10^(n+1)*10^n/n^(10) `


`=lim_(n->oo)(n+1)^10/n^(10)*lim_(n->oo)10^n/10^(n+1) `


`=1*1/10=1/10 `


Since this is less than 1, the series converges. (The sum is a little more than 376.)

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