Apply the ratio test and state its conclusion: ā n=1, ā (n)^100/(100)^n

XApply the ratio test and state its conclusion:

Apply the ratio test and state its conclusion: ā n=1, ā (n)^100/(100)^n

XApply the ratio test and state its conclusion:

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New Wilmington, PA

We need to look at the ratio of successive terms, a_{n+1}/a_{n}, and check if in the limit nāā the ratio is > or < than 1:

((n+1)^{100}/100^{n+1})/(n^{100}/100^{n}) = ((n+1)^{100}/n^{100})/(100^{n+1}/100^{n}) = (1+1/n)^{100}/100

The limit is 1/100, which is less than 1, therefore by the ratio test, the series converges.

Conceptually: the exponential function in the denominator increases much faster than the power function in the numerator, so their ratio gets small fast.

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