Sum of i^2/(4^i): 0 to Infinity

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Homework Help Overview

The discussion revolves around the summation of the series \( \sum_{i=0}^{\infty} \frac{i^2}{4^i} \). Participants are exploring the convergence of this series and considering various approaches to approximate or analyze it.

Discussion Character

  • Exploratory, Assumption checking, Mixed

Approaches and Questions Raised

  • Participants are discussing the convergence of the series and whether certain approximations can be made. There are references to the sequence of partial sums and their properties. Some participants question the appropriateness of the problem's complexity for the intended audience.

Discussion Status

The conversation includes hints and suggestions for approximating the series, with some participants sharing their experiences in calculating partial sums. There is a mix of perspectives on the problem's difficulty, and while some guidance has been offered, there is no explicit consensus on a single approach.

Contextual Notes

There are mentions of the problem being categorized under "K-12 homework," which raises questions about its suitability for that level. Additionally, some participants express interest in exploring less advanced methods for solving the series.

Johnny Leong
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Please give me some hints:

Sum of (i^2)/(4^i) where i is from 0 to infinity.
 
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If you just want an approximate answer, then consider the convergence of the series.
 
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?
 
Well, that was what you were told when you posted this under "k-12 homework".
 
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.
 
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
 
cookiemonster said:
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..
 
  • #10
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.




arildno said:
So would I!
Perhaps I shot a sparrow with a cannon..
I would say you fought an armored knight with a rapier.
 

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