Proving $\frac{1}{n^2}$ infinite series converges without integral test [duplicate]
Just out of curiosity, I was wondering if anybody knows any methods (other than the integral test) of proving the infinite series where the nth term is given by $\frac{1}{n^2}$ converges.
Solution 1:
Hint: $$\frac{1}{n^2} < \frac{1}{n(n-1)}.$$
$$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{n(n - 1)} = 1 - \frac12 + \frac12 - \frac13 + \ldots + \frac{1}{n - 1}-\frac1n = 1 - \frac1n \to 1.$$
Solution 2:
With fewer words. Hopefully clear enough. Oresme's style, but converging this time, and proving that the sum is $<2$.
