Computing a Geometric Power Series with Cosine Terms

In summary, the conversation is about finding the sum of the series \sum_{n=0}^{\infty} p^n cos(3nx) for \abs{p} \textless 1, where p \in \mathbb{R}. The question is whether this can be approached as a telescoping series or if there is a more expedient method. One suggestion is to rearrange using the fact that cos(3nx) = 1/2(exp(3nix)-exp(-3inx)) and then work with the series \sum_{n=0}^{\infty} (pe^{3xi})^n + \sum_{n=0}^\infty (pe^{-3
  • #1
quasar_4
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Homework Statement



Compute [tex]\sum_{n=0}^{\infty} p^n cos(3nx)[/tex] for [tex] \abs{p} \textless 1 [/tex], where [tex] p \in \mathbb{R} [/tex].

Homework Equations





The Attempt at a Solution



I was thinking that maybe this could be approached as a telescoping series, but I'm not really sure if it is. Would that be the most expedient approach? Clearly it isn't geometric, and I'm not sure how to find the sum of a general power series.
 
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  • #2
No one? Someone has to know. There are a lot of clever people on this forum.
 
  • #3
What if I rearrange it using the fact that cos(3nx) = 1/2(exp(3nix)-exp(-3inx)). Could I then write [tex] \sum_{n=0}^{\infty} (pe^{3xi})^n + \sum_{n=0}^\infty (pe^{-3ix})^n [/tex] and try to work from there? (ie, is that valid?)
 
  • #4
Ok, I was being silly, as usual - it is actually geometric after all.

Thank you to me for figuring out this problem.
 

FAQ: Computing a Geometric Power Series with Cosine Terms

What is a power series?

A power series is a mathematical representation of a function where the terms of the series are powers of a variable. It is usually written in the form of an infinite sum, where each term is a constant multiplied by the variable raised to a power.

How do you compute a power series?

To compute a power series, you need to find the general term of the series, which is the pattern that each term follows. Then, you can use this general term to write out as many terms as needed. Finally, you can use mathematical techniques such as substitution, integration, or differentiation to manipulate the series and find its sum.

What are some applications of power series?

Power series have various applications in mathematics, physics, engineering, and other fields. They can be used to approximate functions, solve differential equations, and study the behavior of complex systems. Some specific examples include Taylor series in calculus, Fourier series in signal processing, and Maclaurin series in physics.

How do you determine the convergence of a power series?

The convergence of a power series depends on the values of the variable and the coefficients of the series. The ratio test and the root test are commonly used techniques to determine convergence. If the limit of these tests is less than 1, the series is absolutely convergent. If the limit is greater than 1 or infinite, the series is divergent. If the limit is equal to 1, further analysis is needed to determine the convergence.

Are there any real-life examples of power series?

Yes, there are many real-life examples of power series. For instance, the value of pi can be approximated using the power series expansion of the arctangent function. The motion of a pendulum can be described using a power series solution to a differential equation. Power series are also used in financial calculations, such as compound interest. Additionally, power series are used in computer algorithms to calculate values of functions to a high degree of accuracy.

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