Sum of i²/4^i from 0 to infinity

[Johnny Leong]

Please give me some hints:

Sum of (i^2)/(4^i) where i is from 0 to infinity.

 

[turin]

If you just want an approximate answer, then consider the convergence of the series.

 

[Johnny Leong]

Sum of (i^2)/(4^i) > 1/4 + 1/4 + 9/16 + Sum of 1/(4^i) where 4<=i<=infinity.
Can this be a good approximation?

 

[HallsofIvy]

Well, that was what you were told when you posted this under "k-12 homework".

 

[turin]

The sequence of partial sums is monotonic, so you definitely know that the infinite series must be greater than any partial series. Then, you know that any number greater than 1 is greater than 1, so that remaining sum also fits the bill.

I actually worked this summation out to about 20 terms, and it appears as though it is a recognizable fraction.

 

[cookiemonster]

arildno solved it quite cleverly in Johnny's duplicate thread (bad Johhny!).

https://www.physicsforums.com/showthread.php?t=31966

cookiemonster

 

[turin]

arildno solved it quite cleverly in Johnny's duplicate thread (bad Johhny!).

https://www.physicsforums.com/showthread.php?t=31966

Perhaps, but somewhat inappropriate for K-12 IMO. Or are they teaching that sort of thing in HS now?

 

[cookiemonster]

I had no trouble following it and I'm just out of high school.

Alternatively, do you know of another, less advanced way of solving it? I'd be interested in seeing it.

cookiemonster

 

[arildno]

I had no trouble following it and I'm just out of high school.

Alternatively, do you know of another, less advanced way of solving it? I'd be interested in seeing it.

cookiemonster

So would I!
Perhaps I shot a sparrow with a cannon..

 

[turin]

Add about twenty of the terms. You can definitely see the series converge. This is "less advanced" (and more straightforward). Though I certainly admit that it does not smack of the elegance provided by arildno.

So would I!
Perhaps I shot a sparrow with a cannon..

I would say you fought an armored knight with a rapier.