Why Does Differentiating a Geometric Series Lead to an Alternating Series?

In summary, to find a power series representation for f(x) = 1/ (1+x)^2, we can use differentiation and the geometric series sum formula. By substituting -x for x, we can remember that the denominator in the formula for the geometric series is 1-x, not 1+x. This results in an alternating series, consistent with the expected result.
  • #1
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Homework Statement


Use differentiation to find a power series representation for f(x) = 1/ (1+x)^2

Homework Equations


geometric series sum = 1/(1+x)

The Attempt at a Solution


(1) I see that the function they gave is the derivative of 1/(1+x).
(2) Therefore, (-1)*(d/dx)summation(x^n) = -1/(1+x)^2
(3) Differentiating the summation gives:
(-1)*[summation (n)x^(n-1)]

However, the book is telling me that for my second step (2) I should be getting
d/dx [summation (-1)^n (x^n)].

Why is it becoming an alternating series here?
 
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  • #2
Remember:

[tex] \frac{1}{1-x} = 1+x+x^2+... [/tex]

so that, after substituting -x for x:

[tex] \frac{1}{1+x} = 1 - x + x^2 + ... [/tex]

You can remember the denominator in the first equation is 1-x by multiplying both sides by (1-x), giving:

[tex] 1 = (1-x)(1+x+x^2+...) = 1 + x + x^2 + ... - x - x^2 -x^3 - ... = 1[/tex]

which is consistent, as opposed to what you'd get if you assume 1+x+... was 1/(1+x). (By the way, these manipulations of infinite sums aren't strictly valid, but they can be made more rigorous by restricting to finite sums and taking a limit at the end).
 
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Related to Why Does Differentiating a Geometric Series Lead to an Alternating Series?

1. What is a power series representation?

A power series representation is an infinite series of the form ∑ cn(x-a)n, where a is a constant and cn are coefficients. It is used to represent a function as an infinite sum of powers of a variable x.

2. How is a function represented using a power series?

A function is represented using a power series by finding the coefficients cn and the center a. The coefficients can be found using various methods such as Taylor series or Maclaurin series. The center a is the point around which the function is expanded.

3. What is the importance of power series representation in mathematics?

Power series representation is important in mathematics as it allows us to approximate functions and perform calculations that would otherwise be difficult. It also helps in understanding the behavior of a function and making predictions about its values.

4. Can all functions be represented using a power series?

No, not all functions can be represented using a power series. Some functions may not have a power series representation, while others may have a power series representation only in a certain interval.

5. How can a power series representation be used to find the value of a function at a specific point?

A power series representation can be used to find the value of a function at a specific point by substituting the value of the variable x in the power series. This will give an approximation of the function's value at that point, which becomes more accurate as more terms in the series are considered.

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