Is there a closed formula for this summation?
I have the summation
$$\sum\limits_{i=1}^n \frac{1}{i^2}$$
And I don't know how to find a closed formula for it. Any ideas?
Solution 1:
The infinite series $$\sum_{i=1}^\infty \frac{1}{i^2}$$ has a closed form. But the finite series does not.
Sometimes we may write it using the "trigamma" function, but this is not much beyond the definition. $$ \sum_{i=1}^n \frac{1}{i^2} = \frac{\pi^2}{6} - \psi^{(1)}(n+1) $$