Let $k\leq n$, I am looking to truncate this sum (with a Big O or Little o) in order to compute just few terms $$ \sum^k_{i=1} (k-(i-1))^{p}\sqrt{y_i} $$ where $y=(y_i)_{i\leq n}$ is a bounded sequence, and $p\in(-\frac{1}{2},+\frac{1}{2})$.
I do not think it is possible to cut a part of this sum, since for $p$ in the above range, the sum does not converge.
I need to speed up an algorithm in order to compute a part of this sum (other ideas are welcome).
I think the idea start from the harmonic/p-series for $p\in (-\frac{1}{2},0)$.
In this case I think , $\int^n_1 x^p dx=(p+1)(n^{p+1}-1) \leq \sum^n_{i=1} i^p$