# Feeling like a tool for not remembering this series.

1. Oct 13, 2012

### trap101

So I'm trying to find the value of this finite series and I'm feeling like a tool for not being able to remember the way to calculate the result...

S = 1/(1+x) + 2/(1+x)2 + 3/(1+x)3 +........+ 10/(1+x)10

I know it's a geometric series, but I cannot figure out what to use as my initial term.

Thanks.

2. Oct 13, 2012

### tiny-tim

hi trap101!

no, it's not geometric (because of the 1 2 3 …)

try differentiating or integrating it

3. Oct 13, 2012

### SammyS

Staff Emeritus
Yes. Integrate the general term of the series, $\displaystyle \frac{k}{(1+x)^k}= k(1+x)^{-k}\ .$

4. Oct 13, 2012

### trap101

Well that's not good because it was actually just a portion of a larger problem I was working on. Observe #5 in the attachment:

So essentially I had done some manipualtions to the point that I obtained:

C/PV0 [ the series from above] + F/(1+x)10

where C/PV, F are constants.

I followed your suggestion of integrating the general term and got:

k(1+x)1-k/(1-k).........but how does this help in obtaining a sum for that series?

5. Oct 13, 2012

### trap101

sorry here was the attachment..

#### Attached Files:

• ###### hw1.pdf
File size:
96.8 KB
Views:
960
6. Oct 13, 2012

### trap101

sorry here was the attachment..

7. Oct 14, 2012

### tiny-tim

hi trap101!
it doesn't, the k/(1-k) messes it up!

you need to fiddle about with the original series to get something that integrates to a straightforward factor-less ∑ (1+x)k

8. Oct 14, 2012

### sankalpmittal

No , tiny-tim !! No need to differentiate or integrate it !

trap101 , the series is not geometric or arithmetic , but it is "Arithmetico-Geometric series" , and there is a way of solving it.

S = 1/(1+x) + 2/(1+x)2 + 3/(1+x)3 +........+ 10/(1+x)10 ..... equation 1

Multiply both the sides of the equation by the common ratio , 1/(1+x) and get another equation.

Arithmetico-Geometric series are of form :

a, (a+d)r , (a+2d)r2 , ......... , {a+(n-1)d} rn-1