How can I find The Big Oh bounds for a summation with multiple variables?

Question

I have this as a homework problem so I won't post the same thing. I'll just post what I need to know to move forward.

$$ \sum_{i=0}^n 10^i i^2 $$

I'd just like to know how to split this summation. whether we need to split it? or just work on the big oh bounds for each individually.

I just want to know how to proceed if two terms within a summation cannot be simplified any further and both are dependent on i.

Answer 1

Since $n$ is a constant with respect to $i$, we can just bring it out of the summation: $$ \sum_{i=0}^n 10^in^2 = n^2\sum_{i=0}^n 10^i = n^2 \left( \frac{10^{n+1} - 1}{9} \right) $$

Answer 2

Hint: Start from the fact that $\displaystyle\sum_{i=0}^na^i=\frac{a^{n+1}-1}{a-1}$. Now, differentiate both sides with regard to a, and then multiply with a. Then repeat the process once again.