currently I am working on a Formular which I want to maximize. I tried to simplify the function with Wolfram. The Result was the following:
$f(k)=\sum_{i=k}^{n} \left( \frac{k-1}{(i-1)n} + \frac{(k-1)x}{(i-1)i} \right) = \frac{x(-k+n+1) + (k-1)\psi^{(0)}(n) – (k-1)\psi^{(0)}(k-1)}{n}$
With $k,n \in \mathbb{N}, x \in \mathbb{R}, 1 \leq k \leq n$ and $0 \leq x \leq 1$
Actually I am not that deep into mathematics and never heard of the gamma or digamma function before. I looked up the Definition and some other stuff regarding to it, but nothing from which I could recreate this transformation.
Is there a trick to transform the function as given?
Best Answer
May be, you will feel more conformable with harmonic numbers since $$\psi ^{(0)}(p)=H_{p-1}-\gamma$$ and
$$\sum_{i=k}^{n} \left( \frac{k-1}{(i-1)n} + \frac{(k-1)x}{(i-1)i} \right)=\frac{k-1}n\sum_{i=k}^{n}\frac 1 {i-1}+(k-1)x\sum_{i=k}^{n}\frac 1 {i(i-1)}$$ On the other hand, by telescoping, $$\sum_{i=k}^{n}\frac 1 {i(i-1)}=\sum_{i=k}^{n} \left(\frac 1 {i-1}-\frac 1 i\right)=\frac{1}{k-1}-\frac{1}{n}$$
So $$f(k)=\frac 1 n \Big(x (n-k+1)+(k-1)\left(H_{n-1}-H_{k-2}\right) \Big)$$ which you wish to maximize (with respect to $k$ ?)