Hi everybody,
$\displaystyle U_n=1+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+.. .+\frac{1}{n^3}$
I must show that $\displaystyle (\forall n \in \mathbb{N}^*) U_n\le 2-\frac{1}{n}$
I used recurrence but i didn't find anything.
Use induction:
The base case $\displaystyle n=1$ is true.
Assume true for $\displaystyle n=k$, then
$\displaystyle U_{k+1}=U_k+\frac{1}{(k+1)^3}\le 1-\frac{1}{k}+\frac{1}{(k+1)^3}$
and with a bit of work this can be manipulated to show that:
$\displaystyle U_{k+1}\le 2 - \frac{1}{k+1}$
which allows one to claim proof bt induction
CB