Numerical result using summation limit

Question

I'm exploring the possibilities of the Limit function:

Example 1:

$\lim_{x\to \infty } \, \frac{\sqrt{x-1}}{x}$

Limit[Sqrt[x - 1]/x, x -> Infinity]

0

Example 2:

$\lim_{n\to \infty } \, \frac{(n!)^{1/n}}{n}$

Limit[n!^(1/n)/n, n -> Infinity]

$1/E$

But when I tried to use a summation limit it did not work out:

$\lim_{n\to \infty } \, \left(\sum _{a=1}^n a^{n-a}\right)$

Limit[Sum[a^(n - a), {a, 1, n}], n -> Infinity]

The above idea was to describe this summation:

$1^{10-1}+2^{10-2}+3^{10-3}+4^{10-4}+5^{10-5}+6^{10-6}+7^{10-7}+8^{10-8}+9^{10-9}+10^{10-10}+\text{...}+\infty ^{10-\infty }$

Is it possible to get some numerical result? Maybe using NSolve, Solve ....

Answer 1

First, we note all the terms of the sum under cosideration are positive. Second, we take one of them, namely

Floor[n/2]^(n - Floor[n/2])

, and consider its limit as $n$ approaches $\infty$:

Limit[Floor[n/2]^(n - Floor[n/2]), n -> Infinity]

$\infty$.

This implies the limit under consideration is infinite too. Of course, there may exist a generalised sum of the series under consideration, but this is a math stuff, not a Mathematica matter.