# Summation of series

1. Feb 7, 2013

### tyneoh

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

Let v1, v2, v3 be a sequence and let

un=nvn-(n+1)vn+1

for n= 1,2,3.... find $\sumun$ from n=1 to N.
2. Relevant equations

3. The attempt at a solution
Began with method of differences and arrived at
Sn= v1-(n+1)vn+1

2. Feb 7, 2013

### Staff: Mentor

I'd first try writing out u1, u2, u3 to see if there's some term cancellations that come about when you sompute
sum (un) = u1 + u2 + u3 + ...

3. Feb 8, 2013

### SammyS

Staff Emeritus

4. Feb 8, 2013

### Karnage1993

I think the question is: given the recursive formula $u_n = nv_n - (n+1)v_{n+1}$, find the general formula for the sum of $u_1 + u_2 + ... + u_n$ for any n.

Let's first list out some possibilities:

$u_1 = v_1 - 2v_2\\ u_2 = 2v_2 - 3v_3\\ u_3 = 3v_3 - 4v_4$

So the sum of the three is:

sum{$u_3$} $= v_1 - 2v_2 + 2v_2 - 3v_3 + 3v_3 - 4v_4 = v_1 - 4v_4$

Based on this, can you think of a formula for any $n$th sum?

Last edited: Feb 8, 2013
5. Feb 9, 2013

### SammyS

Staff Emeritus
If you mean $\displaystyle\ \ S_N=\sum_{n=1}^{N}u_n=v_1-(N+1)v_{N+1},,\$ then your result looks good.

Last edited: Feb 9, 2013
6. Feb 9, 2013

### Staff: Mentor

Do you have one too many equals?

7. Feb 9, 2013

### SammyS

Staff Emeritus
LOL !

Thanks!

Actually I had one too few un .

I'll edit my post!