# Is the series #\sum_(n=0)^\infty n^n/(4^n n!)# absolutely convergent, conditionally convergent or divergent?

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(Use the appropriate test.)

(Use the appropriate test.)

##### 1 Answer

Converges absolutely by the Ratio Test.

#### Explanation:

To determine absolute convergence, we use the Ratio Test, which tells us if we have some series

Here,

Then,

Recalling that division is multiplication by the reciprocal:

We've dropped the absolute value bars -- as we go to infinity, everything is positive, so no need for absolute value.

Furthermore, that limit is

We could prove this with l'Hospital's Rule, but differentiating the bottom is extremely tedious, so the geometric argument is fine.

Thus, the series converges absolutely.