Question

What is the interval of convergence of `sum (n^n)(x^n)`?

Answer 1

`0`

Explanation:

If `abs(x) > 0` then for any `n > 1/(abs(x))` we have:

`EE epsilon > 0 : n > (1+epsilon)/(abs(x))`

So:

`abs(n^n x^n) > ((1+epsilon)/(abs(x)) abs(x))^n = (1+epsilon)^n >= 1+n epsilon`

So:

`lim_(n->oo) abs(n^n x^n) = oo`

So the terms of the series do not converge to `0`, let alone the sum, unless `x = 0`.

Answer 2

See below.

Explanation:

If the series converges then

`(n absx)^n ge ((n+1)absx)^(n+1)` or

`(n/(n+1))^n ge (n+1)absx`

but `(n/(n+1))^n lt 1` so if

`lim_(n->oo) (n+1)absx < 1 rArr abs x = 0`