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I am trying to solve the following limit as a part of using the n-th test for convergence:

$\lim_{n \to \infty}(\frac{n}{7})^n$

Is it possible to massage this expression further at all (analytically), or are we just supposed to know that this approaches infinity?

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For every $n\geqslant14$, one has $\frac{n}{7}\geqslant2$ hence $\left(\frac{n}{7}\right)^n\geqslant2^n$, and this should allow you to conclude readily.

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