Good evening,
Let $n, b \in \mathbb{N}$, $n,b \geq 2$ and $b \in \mathbb{N}$, then we know that
$\sum_{k=0}^n b^k = \frac{b^{n+1}-1}{b-1}$.
In other words we have $\sum_{k=0}^n b^k = c_{1,b} \cdot b^n + c_{2,b}$, where $c_{1,b},c_{2,b}$ are constants only depending on $b$.
I'm wondering, if we can find similar expressions, if we consider a non-decreasing sequence of positives integers, say $a_k$.
In other words, I'm looking for an expression
$\sum_{k=0}^n \frac{b^k}{a_k} = c'_{1,b} \cdot \frac{b^n}{a_n} + c'_{2,b}$. Is this in general possible? I tried a little bit with $a_k = \log(k+2)$, but I couldn't even come up with an estimate of the type $\sum_{k=0}^n \frac{b^k}{\log(k+2)} \leq c_1 \cdot \frac{b^n}{\log(n+2)} + c_2$.
Does anyone out there have any suggestions for an approach?
Thanks in advance.
No. Consider $a_k=d^k$. Then it is again the geometric series with a partial sum depending also on $d$.