What is the process for solving for the truncation error for a series? Give an appropriate upper bound for the following truncation error:
$$\left| \sum_{n=1}^{\infty} \frac{1}{n^2} - \sum_{n=1}^{100} \frac{1}{n^{2}} \right|$$
Please explain the process for solving for the upper bound truncation.
Thank you

The error is bounded by
$$\begin{align}\sum_{n=101}^\infty\frac1{n^2} 
&< \sum_{n=101}^\infty\frac1{n^2-n}\\
&=  \sum_{n=101}^\infty\left(\frac1{n-1}-\frac1n\right)\\
&=\lim_{N\to\infty}\left(\sum_{n=101}^N\left(\frac1{n-1}-\frac1n\right)\right)\\
&=\lim_{N\to\infty}\left(\sum_{n=101}^N\frac1{n-1}-\sum_{n=101}^N\frac1{n}\right)\\
&=\lim_{N\to\infty}\left(\sum_{n=100}^{N-1}\frac1{n}-\sum_{n=101}^N\frac1{n}\right)\\
&=\lim_{N\to\infty}\left(\frac1{100}-\frac1N\right)\\&=\frac1{100}.
\end{align}$$
We "wasted" something only in the first step - but was it much?
It turns out it wasn't, for we also have
$$\sum_{n=101}^\infty\frac1{n^2} >\sum_{n=101}^\infty\frac1{n^2+n}=\frac1{101}. $$
