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

[Johnny Leong]

Please give me some hints:

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

 

[NateTG]

Please give me some hints:

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

You mean \sum_{i=1}^\infty \frac{i^2}{4^i}? (I don't like \frac{0^2}{4^0} - it's not necessarily clear what it is.)

Well, for i \geq 4 ,
2^i \geq i^2,
so
\frac{i^2}{4^i} < \frac{2^i}{4^i} = \frac{1}{2^i}
so
\sum_{i=1}^\infty \frac{i^2}{4^i} < \frac{1}{4}+\frac{4}{16}+\frac{9}{64} + \sum_{i=4}^{\infty} \frac{1}{2^i}

Perhaps you can find some better bounds?

 

[Muzza]

I'm sorry, but what's unclear about 0^2/4^0?

 

[Gokul43201]

\sum_{i=1}^\infty \frac{i^2}{4^i} > \frac{1}{4}+\frac{4}{16}+\frac{9}{64} + \sum_{i=4}^{\infty} \frac{1}{4^i}

That gives a range of 0.7 to 0.9. Wonder if it isn't just 3/4 or 4/5 ?

 

[cookiemonster]

Mathematica's saying 20/27 (~.740741). But, like most things Mathematica, I have no idea how it produced that number.

cookiemonster

 

[Johnny Leong]

Actually, this question could not give an accurate answer. The answer should just be an approximation, right? Because the terms in the summation are not having some sequence properties.

 

[Johnny Leong]

I think Gokul43201's answer would not be a good one because after I have read a book, I found that when we do some approximation, the approximation's value had better dominate the value of the original question.

 

[HallsofIvy]

I think Gokul43201's answer would not be a good one because after I have read a book, I found that when we do some approximation, the approximation's value had better dominate the value of the original question.

What in the world does that mean? What do you mean by "dominate the value of the original question? An approximation is an approximation. There exist good approximations and bad approximations. The ideal is to get as close as possible to the true value for the work done.

 

[arildno]

F(x)=\sum_{i=0}^{\infty}i^{2}x^{i}
\sum_{i=0}^{\infty}i^{2}x^{i}=\sum_{i=0}^{\infty}xi\frac{d}{dx}x^{i}
\sum_{i=0}^{\infty}xi\frac{d}{dx}x^{i}=\frac{d}{dx}\sum_{i=0}^{\infty}ix^{i+1}-\sum_{i=0}^{\infty}ix^{i}
\sum_{i=0}^{\infty}ix^{i}=x\frac{d}{dx}\sum_{i=0}^{\infty}x^{i}
Hence, we have:
F(x)=\frac{d}{dx}(x^{2}\frac{d}{dx}\frac{1}{1-x})-x\frac{d}{dx}\frac{1}{1-x}
or:
F(x)=\frac{x^{2}+x}{(1-x)^{3}}
The sum of the original series is found by evaluating F(\frac{1}{4})

 

[Gokul43201]

Which is 20/27, as Mathematica magically guessed.

Nice, arildno.

 

[cookiemonster]

A very clever little trick, indeed.

cookiemonster

 

[Johnny Leong]

What in the world does that mean? What do you mean by "dominate the value of the original question? An approximation is an approximation. There exist good approximations and bad approximations. The ideal is to get as close as possible to the true value for the work done.

I mean the approximation should be an upper bound to the accurate answer to the original question.

 

[Johnny Leong]

F(x)=\sum_{i=0}^{\infty}i^{2}x^{i}
\sum_{i=0}^{\infty}i^{2}x^{i}=\sum_{i=0}^{\infty}xi\frac{d}{dx}x^{i}
\sum_{i=0}^{\infty}xi\frac{d}{dx}x^{i}=\frac{d}{dx}\sum_{i=0}^{\infty}ix^{i+1}-\sum_{i=0}^{\infty}ix^{i}
\sum_{i=0}^{\infty}ix^{i}=x\frac{d}{dx}\sum_{i=0}^{\infty}x^{i}
Hence, we have:
F(x)=\frac{d}{dx}(x^{2}\frac{d}{dx}\frac{1}{1-x})-x\frac{d}{dx}\frac{1}{1-x}
or:
F(x)=\frac{x^{2}+x}{(1-x)^{3}}
The sum of the original series is found by evaluating F(\frac{1}{4})

What are you doing, arildno? You are professional but I do not understand. Why you do like this?

 

[arildno]

OK, I'll break up this in tiny pieces; then pin-point what you don't understand.

1. Changing perspective from number to function:
We start out with the series:
S=\sum_{i=0}^{\infty}i^{2}(\frac{1}{4})^{i}
Our aim is to find the number S!
However, this is difficult to do as it stands; what I want to do, is to change the problem slightly, so that:
a) I gain access to powerful solving techniques in the new problem to be solved
(which is not accessible in the original problem)
b) I can easily find the answer to the original problem once I have found the answer to the new problem

That is why I change perspectives to try to simplify the expression for the following function:
F(x)=\sum_{i=0}^{\infty}i^{2}x^{i}

a): Since I now have a power series function , I can use the extraordinarily powerful teqnique of differentiation to help me solve the new problem (simplifying the expression for F(x)!)
b) If I can do this, then I can find the answer to my original problem simply by
computing F(\frac{1}{4})

I'll let you ponder on this for a while, if you have some specific questions, pleasy notify..