Textbook question on power series

In summary, the example in the textbook has an equation for the coefficients of a power series and shows how to get n+1 from it by multiplying out the power series.
  • #1
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My textbook has an example on multiplication of power series.
" Multiply the geometric series [tex] x^n[/tex] by itself to get a power series for [tex]1/(1-x)^2 [/tex] for |x|<1 "
from this we get the [tex]c_{n}[/tex]=n+1
O.K. I get that the coefficients are 1 for all n but why +1.
Could someone please explain this to me if possible.
 
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  • #2
You want to multiply [tex] (1 + x + x^2 + x^3 + ...)(1 + x + x^2 + x^3 + ...)[/tex], so just try few examples first to see what's going on. The [tex]x^0[/tex] term is just [tex] 1*1 = 1[/tex], the [tex] x^1 [/tex] term is [tex] x*1+1*x = 2x [/tex], the [tex] x^2 [/tex] term is [tex] x^2*1 + x*x + 1*x^2 = 3x^2 [/tex], and the [tex] x^3 [/tex] is [tex] x^3*1 + x^2*x+x*x^2+1*x^3 =4x^3[/tex]. Now, can you begin to see a pattern forming?

Alternatively, you can just apply the formula for the coeffecients of the product series in terms of the coeffecients of the two original series. It looks something like [tex] c_k = \sum^k_{r=0} a_r b_{k-r}[/tex] where the a's and b's are the coeffecients of the original series, and the c's are coeffecients of the product series.
 
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  • #3
Right call Physics Monkey:
The formula for the Cauchy product of series as it is presented in this http://mwt.e-technik.uni-ulm.de/world/lehre/basic_mathematics/di/node14.php3 [Broken].
A quick version is:
Suppose [itex] \sum_{n=0}^{\infty} a_n[/itex] and [itex] \sum_{n=0}^{\infty} b_n[/itex] converge absolutely. Then
[tex]\left( \sum_{n=0}^{\infty} a_n\right) \left( \sum_{n=0}^{\infty} b_n\right) = \sum_{n=0}^{\infty} \sum_{k=0}^{n} a_{k}b_{n-k}[/tex] also converges absolutely.
Alternately, look here, under the heading A Variant.
 
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  • #4
O.K. I see what's going on now, thanks. But I still don't get how to get n+1 from [tex] c_k = \sum^k_{r=0} a_r b_{k-r}[/tex] without doing some multiplication. Please explain to me if you can.
 
  • #5
the answer to you above question

Using

[tex]\left( \sum_{n=0}^{\infty} a_n\right) \left( \sum_{n=0}^{\infty} b_n\right) = \sum_{n=0}^{\infty} \sum_{k=0}^{n} a_{k}b_{n-k}[/tex]

we have

[tex]\left( \sum_{n=0}^{\infty} x^n\right) \left( \sum_{n=0}^{\infty} x^n\right) = \sum_{n=0}^{\infty} \sum_{k=0}^{n} x^{k}x^{n-k} = \sum_{n=0}^{\infty} \sum_{k=0}^{n} x^{k+(n-k)} = \sum_{n=0}^{\infty} x^{n} \sum_{k=0}^{n} 1 = \sum_{n=0}^{\infty}(n+1) x^{n} [/tex]

where [itex]\sum_{k=0}^{n} 1 =n+1[/itex] is the answer to you above question.
 
  • #6
O.K. I getting close to understanding this. What puzzles me is the last sum. Do you put n+1 so the first term isn't 0? I think I'm missing some information to understand this.
 
  • #7
We put n+1 because that is what it is... You're adding up the number 1, n+1 times, so the answer is n+1.

But what's wrong with just multiplying out the power series?You can multiply (1+x+x^2+...)(1+x+x^2+...)

and count and think and, well, it's just true... there is nothing clever going on. to end up with x^n in the product you can only get it from multiplying x^r in the first and x^{n-r} and each of those multiplications contributes 1 to the coefficiant of x^n and there is one contributrion from each r as r goes from 0 to n so you get 1 added up n+1 times.
 
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  • #8
I think I understand this no, just hvae to get some practice solving these kind of problems. Thanks.
 

1. What is a power series?

A power series is an infinite series where each term is a constant multiplied by a variable raised to a non-negative integer power. It can be written in the form of ∑n=0∞ cnxn, where cn is the coefficient and x is the variable.

2. What are some examples of power series?

Some examples of power series include the geometric series (1 + x + x2 + x3 + ...), the exponential series (1 + x + x2/2! + x3/3! + ...), and the sine and cosine series (sin(x) = x - x3/3! + x5/5! - ... and cos(x) = 1 - x2/2! + x4/4! - ...).

3. What is the interval of convergence for a power series?

The interval of convergence for a power series is the range of values for the variable x that will result in the series converging, or approaching a finite value. It can be determined using the ratio test or the root test.

4. How can power series be used to approximate functions?

Power series can be used to approximate functions by finding the Taylor series expansion of the function, which is a power series representation that is centered at a specific point. By using more terms in the series, the approximation becomes more accurate.

5. What is the purpose of using a power series in math and science?

Power series are useful in math and science because they can represent a wide range of functions and can be used to approximate these functions with varying levels of accuracy. They are also important in calculus and differential equations, as they allow for the integration and differentiation of more complex functions.

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