Zeta Function Doesn't Matter

Relevant wiki: Stolz–Cesàro theorem

$\begin{aligned} M&=\lim_{n\to\infty}n^2\left[A-\left(1+\frac1{2^3}+\frac1{3^3}+\frac1{4^3}+\cdots+\frac1{n^3}\right)\right] \\&=\lim_{n\to\infty}\frac {A-\left(1+\frac1{2^3}+\frac1{3^3}+\frac1{4^3}+\cdots+\frac1{n^3}\right)}{\frac 1{n^2}}&\small \color{#3D99F6}\text{By Stolz-Cesàro theorem} \\&=\lim_{n\to\infty}\frac {-\frac 1{(n+1)^3}}{\frac 1{(n+1)^2}-\frac 1{n^2}} \\&=\lim_{n\to\infty}\frac {n^2}{(2n+1)(n+1)} \\&=\lim_{n\to\infty}\frac 1{\left(2+\frac 1n\right)\left(1+\frac 1n\right)} \\&=\frac 12=\boxed{0.5}. \end{aligned}$

The sum $\sum \frac1{k^3}$ converges by standard tests, and it clearly must converge to $A$ for the limit to exist. Then $\begin{aligned} M &= \lim_{n\to\infty} n^2 \left( \frac1{(n+1)^3} + \frac1{(n+2)^3} + \cdots \right) \\ &= \lim_{n\to\infty} \frac1{n} \left( \frac1{(1+\frac1{n})^3} + \frac1{(1+\frac2{n})^3} + \cdots \right) \\ &= \int\limits_1^\infty \frac1{x^3} \, dx \\ &= -\frac1{2x^2} \bigg|_1^\infty \\ &= \fbox{0.5}. \end{aligned}$