# How can i express this Infinite series without a summation symbol?

bobthebanana
(1/2) + (2/4) + ... + (n/(2^n))

=

sum i=1 to i=infinity of (i/(2^i))?

i know how to express the sum of just 1/(2^i), but not the above

thanks for the help!

Homework Helper
Gold Member
Dearly Missed
I don't understand the question.
1. Do you want a symbolic notation for a sum without using the standard summation symbol?
2. Are you instead asking for how the standard summation symbol looks like?
3. Do you wish to find an alternate expression for the sum, i.e, calculate it in a manner so that it is easy to evaluate it for an arbitrarily chosen n?

Homework Helper
Gold Member
Dearly Missed
I think you mean 3, i.e, how to calculate that sum. Am I right?

bobthebanana
yea number 3

ice109
$$2^{-n} \left(-n+2^{n+1}-2\right)$$

Pere Callahan
$$2^{-n} \left(-n+2^{n+1}-2\right)$$

This is so instructive...!

exk
the summation is not 0, the above is only slightly correct ;)

ice109
This is so instructive...!

Homework Helper
Gold Member
Dearly Missed
Let us consider the function:
$$F(x)=\sum_{i=1}^{\infty}i*x^{i}$$
Note that F(1/2) equals your sum!
Now, we may write:
$$F(x)=x*\sum_{i=1}^{\infty}i*x^{i-1}=x*\frac{d}{dx}\sum_{i=1}^{\infty}x^{i}=x*\frac{d}{dx}(\frac{1}{1-x}-1)=\frac{x}{(1-x)^{2}}$$
Hence, we easily gain F(1/2)=2.

For arbitrary finite n, use a similar procedure.

LukeD
Bah, I wrote a similar response using generating functions... twice... and physics forums died on me both times so nothing was posted.

Arildno's response is entirely correct though