Feeling like a tool for not remembering this series.

[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.

 

[tiny-tim]

hi trap101!

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

try differentiating or integrating it ​

 

[SammyS]

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.

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

 

[trap101]

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

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?

 

[trap101]

sorry here was the attachment..

 

[trap101]

sorry here was the attachment..

 

[tiny-tim]

hi trap101!

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

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

 

[sankalpmittal]

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

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} r^n-1