- #1
Onezimo Cardoso
Homework Statement
Prove that [itex]\lim_{n \to \infty} \displaystyle \sum_{i=1}^{n} \frac{1}{n+i} = log(2)[/itex].
Homework Equations
[/B]
The Digamma Function [itex]\Psi(x)[/itex] can be written as:
[itex]\Psi (x) = \frac{d}{dx} \log(\Gamma(x))[/itex]
Or, equivalently, as
[itex]\Psi(x)=\displaystyle \sum_{l=1}^{k} \frac{1}{x-l}+\Psi(x-k)[/itex]
The Attempt at a Solution
I tried to put in Wolfram in order to get some hint about this problem and it showed that:
[itex]\displaystyle \sum_{i=1}^{n} \frac{1}{n+i} = \Psi(n+(i+1))-\Psi(1+i)[/itex]
Where [itex]\Psi (x) = \frac{d}{dx} \log(\Gamma(x))[/itex] is the digamma function as described in Relevant equations.