Find upper and lower bound for the following finite sum
$1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$
My attempt:
$1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n 1/(1+i^3)$ = $\int_1^n$1/$(1+i^3)$di = $\int_1^n1/(1+x^3)$dx
But now I'm stuck. What should I do? Is m attempt correct?