Reciprocal over a summation

Question

Is this statement true? Can we take reciprocal over a summation? $$\frac 1{\sum_{n=1}^\infty\frac 1{(n+1)^3}}=\sum_{n=1}^\infty (n+1)^3$$

Answer 1

It is not true. By simple inspection, the sum in the denominator on the left side is a finite number, so the left side is finite. But the right side is apparently infinite.

Answer 2

No. $\sum_{n=1}^\infty \frac{1}{(n+1)^3}=\zeta(3)-1$, so it's reciprocal is $\frac{1}{\zeta(3)-1}$ which is a finite number. The term on the right is obviously divergent.

For reference, $\zeta(3)\approx 1.20205690315959...$ and is called Apery's constant.