By Induction, prove $1^n+2^n+3^n+...+n^n < (n+1)^n$

Question

Here is the question:

Given that $$x^n + n( x^{n-1} ) \leq (x + 1) ^n \tag{A}$$

Prove: $$1^n + 2^n +3^n +...+n^n < (n+1)^n. \tag{B}$$

Here is my thinking.

Replace $n$ with $x$ in A can get $n^n + n(n^{n-1})$ which is $2n^n$

In inductive step, $1^{n+1} + 2^{n+1} + .... + (n+1)^{n+1}$

I separate it into $(1^n +2^n+...+n^n) + (2^n + 2(3^n)+3(4^n) +.....)$

But then I don't know how to do next.

Answer 1

A rough sketch of the induction step for n:

Assume $1^n + \dots + n^n < (n+1)^n$. Our goal is to show that $1^{n+1} + \dots + (n+1)^{n+1} < (n+2)^{n+1}$. So,

\begin{alignat}{2} (n+2)^{n+1} &\geq (n+1)^{n+1} + (n+1)(n+1)^n &\qquad \mbox{(by A)} \\ &> (n+1)^{n+1} + (n+1)(1^n + \dots + n^n) &\\ &\geq (n+1)^{n+1} + 1^{n+1} + \dots + n^{n+1} & \end{alignat}