<s>[INST] Which of the two sums gives a better approximation to $\pi^2/6$? Which of the following two sums 
$$\sum_{n=1}^{1000000}\frac1{n^2} \quad \text{or} \quad 1+\sum_{n=1}^{1000}\frac1{n^2(n+1)}$$
gives a better approximation to $\pi^2/6?$
I tested this on MATLAB and surprisingly obtained as a result the second sum. However, I do not know how to prove this rigorously.
 [/INST] Let: $$G_{10^6}=\sum_{n=1}^{1000000}\frac1{n^2} \quad \text{and} \quad H_{10^3}+1=\sum_{n=1}^{1000}\frac1{n^2(n+1)}+1.$$
We can now compare the remainders of the two and see which is the smallest:
$$\begin{aligned}
&1) \int_{10^6+1}^\infty\frac1{x^2}dx\le\underbrace{\frac{\pi^2}6-G_{10^6}}_{\text{$R_G$}}\le\int_{10^6}^\infty\frac1{x^2}\,dx.\\
&2) \int_{10^3+1}^\infty\frac1{x^2(x+1)}dx\le\underbrace{\frac{\pi^2}6-1-H_{10^3}}_{\text{$R_H$}}\le\int_{10^3}^\infty\frac1{x^2(x+1)}\,dx.\end{aligned}$$
We can evaluate the improper integrals:
$$\begin{aligned}&1)\ \lim_{C\to\infty}\int_N^C\frac1{x^2}dx=\lim_{C\to\infty}\left[-\frac1x\right]^C_N=\frac1N.\\
&2)\ \lim_{S\to\infty}\int_K^S\frac1{x^2(x+1)}dx=\lim_{S\to\infty}\int_K^S\left(\frac1{x^2}-\frac1x+\frac1{x+1}\right)dx=\lim_{S\to\infty}\left[\ln\left(\frac{x+1}x\right)-\frac1x\right]^S_K=\frac1K-\ln\left(\frac{K+1}K\right).\end{aligned}$$
Thus: $$\begin{aligned}&1)\ \frac1{10^6+1}\le R_G\le\frac1{10^6}\Leftrightarrow 9.\overline{9}\cdot10^{-7}\le R_G\le 10^{-6}.\\ &2)\ \frac1{10^3+1}-\ln\left(\frac{10^3+2}{10^3+1}\right)\le R_H\le \frac1{10^3}-\ln\left(\frac{10^3+1}{10^3}\right)\end{aligned}$$ $$\approx4.98\cdot10^{-7}\le R_H \le4.99\cdot10^{-7}.$$
We can clearly see that $R_G>R_H$, therefore $$1+\sum_{n=1}^{1000}\frac1{n^2(n+1)}$$ gives a better approximation to $\pi^2/6$.
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