Limit of Finite Sum: Solving the Convergence for a Series with Zero Sum

[dobry_den]

Homework Statement

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

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

where

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

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.

 

[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.

 

[dobry_den]

Oh yes, thanks a lot!