# Limit of a finite sum

1. Jan 22, 2008

### dobry_den

1. The problem statement, all variables and given/known data

Hey, I'm trying to determine the following limit:

$$\lim_{n \rightarrow \infty}\sum_{j=0}^k{a_j\sqrt{n+j}}$$

where

$$\sum_{j=0}^k{a_j}=0$$

3. The attempt at a solution

I tried to go this way:

$$\lim_{n \rightarrow \infty}\sqrt{n+k}\sum_{j=0}^k{a_j\frac{\sqrt{n+j}}{\sqrt{n+k}}}$$

but still I get 0*infinity, which is undefined.

2. Jan 22, 2008

### Dick

Let M be the sum of all of the positive a_j, so the sum of all the negative a_j is -M. Now split the sum into the sum of all the positive members, call it Sp and the sum of all the negative members, call it Sn. So the whole sum is Sp+Sn. Convince yourself that Sp<=M*sqrt(n+k) and Sn<=(-M)*sqrt(n). So an upper bound for the sum is M(sqrt(n+k)-sqrt(n)). Show that approaches zero. Now find a lower bound and repeat.

3. Jan 22, 2008

### dobry_den

Oh yes, thanks a lot!