Let $a>1$ and consider the following finite series: $$ 1+\frac{2}{a}+\frac{3}{a^2}+\cdots+\frac{n}{a^{n-1}}, $$ where $n\geq 1$ is a fixed quantity.
Then is the above series uniformly bounded by a constant independent of $n$?
I tried to proceed as follows: We know that $$ 1+\frac{2}{a}+\frac{3}{a^2}+\cdots+\frac{n}{a^{n-1}}=\frac{r^{n+1}n+1-(n+1)r^n}{(1-r)^2} $$ where $r=\frac{1}{a}$.
But unable to proved the independency. Can somebody kindly help. Thanks.
If $a > 1$ you get that $nr^n \to 0$, thus $$\lim_{n \to \infty} \Big( 1 + \dots +\frac{n}{a^{n-1}} \Big) = \lim_{n \to \infty} \frac{r^{n+1}n+1-(n+1)r^n}{(1-r)^2} = \frac{1}{(1-r)^2}$$and the sum is bounded by its limit because $$S_n = 1 + \dots + \frac{n}{a^{n-1}}$$ is strictly increasing with respect $n$.