Special Series — Question 1

*Question 1 Consider the Basel series $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2}$.

1. Prove that the series converges using the $p$-series test.

2. State Euler’s classical evaluation of its exact sum and give a decimal approximation.

3. Use an integral estimate to bound the remainder after $N$ terms, and explain why convergence alone does not produce the exact constant.

*Question 1 – Solution

This is a $p$-series $\sum1/n^p$ with $p=2>1$. Therefore it converges. This test establishes existence of a finite sum but does not determine that sum.

Euler’s solution of the Basel problem gives the non-elementary identity $$\sum_{n=1}^{\infty}\frac1{n^2}=\zeta(2)=\frac{\pi^2}{6}\approx1.644934.$$ The equality with $\pi^2/6$ requires additional machinery, such as Euler’s product for $\sin x$ or Fourier series; it does not follow from the $p$-series test alone.

Because $f(x)=1/x^2$ is positive and decreasing, $$\int_{N+1}^{\infty}\frac{dx}{x^2}\le R_N \le\int_N^{\infty}\frac{dx}{x^2},$$ so $$\frac1{N+1}\le R_N\le\frac1N.$$ Thus the partial sums approach $\pi^2/6$ from below, with an error of order $1/N$.