Help with Summation: Evaluate 1/4+2/16+3/64+4/256+5/1024+...

[darkvalentine]

Homework Statement

Evaluate: 1/4+2/16+3/64+4/256+5/1024+...

Homework Equations

The Attempt at a Solution

The summation can be written as: Sum(k=1 to infinity, k/(4^k))
Then I do not know how to calculate the sum. Please help!

 

[Dick]

You know how to sum x^k/(4^k), right? It's a geometric series. It gives you some function f(x). Now consider what f'(x) is evaluated at x=1.

 

[darkvalentine]

You know how to sum x^k/(4^k), right? It's a geometric series. It gives you some function f(x). Now consider what f'(x) is evaluated at x=1.

Honestly I do not know how to sum x^k/(4^k), can you explain a little more why we have to put it in a function f(x)? f'(x) at x=1 going to be (k-ln4)/(4^k) but then ?

 

[Dick]

Honestly I do not know how to sum x^k/(4^k), can you explain a little more why we have to put it in a function f(x)? f'(x) at x=1 going to be (k-ln4)/(4^k) but then ?

The sum of x^k/4^k is geometric because it's the sum of (x/4)^k. Look up the formula for summing a geometric series. The common ratio r=x/4, yes? The result is a function of r, which x/4. So it's a function of x. And when I say f'(x) I mean the derivative with respect to x. Isn't it sum k*x^(k-1)/4^k? No logs needed. Do you see how the k in your sum comes in?

 

[darkvalentine]

The sum of x^k/4^k is geometric because it's the sum of (x/4)^k. Look up the formula for summing a geometric series. The common ratio r=x/4, yes? The result is a function of r, which x/4. So it's a function of x. And when I say f'(x) I mean the derivative with respect to x. Isn't it sum k*x^(k-1)/4^k? No logs needed. Do you see how the k in your sum comes in?

Thanks, I got it ^^