Finding the Sum of a Power Series

In summary, the sum of the given equation is equal to (-1)^n x^n, which can be obtained by manipulating the equation to be in the form of a geometric series. The key is to use the formula for a geometric series, where the common ratio is -x. So, by substituting -x for x in the formula, we get (-1)^n x^n, which is the desired sum.
  • #1
student45
I'm trying to find the sum of this:

[tex]
\[
\sum\limits_{n = 0}^\infty {( - 1)^n nx^n }
\]
[/tex]

This is what I have so far:

[tex]
\[
\begin{array}{l}
\frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {x^n } \\
\frac{1}{{(1 - x)^2 }} = \sum\limits_{n = 0}^\infty {nx^{n - 1} } = \sum\limits_{n = 1}^\infty {nx^{n - 1} } \\
\frac{x}{{(1 - x)^2 }} = \sum\limits_{n = 1}^\infty {nx^n } \\
\end{array}
\]
[/tex]

So how do I get the (-1)^n part in there? Any suggestions would be really helpful. Thanks.
 
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  • #2
[tex] \frac{1}{1+x} = \sum_{n=0}^{\infty} (-1)^{n}x^{n} [/tex]
 
Last edited:
  • #3
Oh, I see.. Where exactly does that come from?
 
  • #4
[tex] \frac{1}{1+x} = \frac{1}{1-(-x)} = \sum_{n=0}^{\infty} (-x)^{n} = (-1)^{n}x^{n} [/tex]
 
  • #5
Ah! Of course. Okay. Thanks.
 

1. What is a power series?

A power series is an infinite series of the form ∑n=0∞ an(x-c)n, where an are the coefficients, x is the variable, and c is the center. It is a type of mathematical series that represents a function.

2. How do you find the sum of a power series?

To find the sum of a power series, you can use the formula for the sum of an infinite geometric series: S = a / (1-r), where a is the first term and r is the common ratio. However, this formula only works if the power series is geometric.

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

The radius of convergence is the distance from the center of the power series where the series converges. It can be calculated using the ratio test, which states that if the limit of |an+1 / an| as n approaches infinity is less than 1, then the power series converges. The radius of convergence is equal to the reciprocal of this limit.

4. Can a power series diverge?

Yes, a power series can diverge if the limit of |an+1 / an| as n approaches infinity is greater than 1. This means that the series does not approach a finite value and the terms continue to increase without bound.

5. How can finding the sum of a power series be useful in science?

Power series are used in many areas of science to approximate functions and make predictions. For example, in physics, power series are used to approximate the behavior of physical systems, such as the motion of a pendulum. In mathematical modeling, power series can be used to approximate complex functions and make calculations more manageable.

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