Why Do Taylor Series Representations of Cosine Use Alternating Powers of -1?

In summary, the conversation discusses the process of developing a series of cosine and representing it as a sum. The power of (-1) is determined by taking only the odd members, but the solution uses an expression of (-1)^{k-1} instead of (-1)^{2k+1}. The reason for this expression is to alternate the signs of the Taylor's series. The speaker confirms that this is correct and asks for instructions on how to obtain this expression.
  • #1
nhrock3
415
0
when i develop the series of a cosine i have a (-1) member
i wanted to represent the series as a sum
so i need to take only the odd members so the power of -1 is 2k+1 i got
but the solution says that the power of -1 is equal (-1)^{k-1}

is it the same??
why they have such an expression
(they use n istead of k)
http://i45.tinypic.com/6sszue.jpg

how they got the power?
 
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  • #2
You want the signs of your Taylor's series to alternate, right? (-1)k - 1 gives you that sign alternation. If you had (-1)2k + 1, the sign would always be negative, since you have odd powers of -1.
 
  • #3
you are correct
how to get this expression?
 
  • #4
What exactly are you asking? Are you asked to find the Taylor's series for cos z at z = 2? There is a standard technique for finding the coefficients of this series.
 

FAQ: Why Do Taylor Series Representations of Cosine Use Alternating Powers of -1?

What is a Taylor series?

A Taylor series is a mathematical representation of a function as an infinite sum of terms, where each term is a function evaluated at a specific point. It is used to approximate the behavior of a function near a given point.

What is the purpose of a Taylor series?

The purpose of a Taylor series is to approximate a function with a polynomial, making it easier to manipulate and analyze. This series allows us to evaluate a function at a specific point without having to explicitly know the function's value at that point.

What is the difference between a Taylor series and a power series?

A Taylor series is a specific type of power series, where the terms of the series are derived by taking derivatives of the function at a specific point. Power series, on the other hand, can have terms that are not dependent on derivatives and can be centered at any point.

What is the convergence of a Taylor series?

The convergence of a Taylor series refers to its ability to accurately approximate a function. A Taylor series will only converge if the function is infinitely differentiable at the point of expansion and the series is evaluated within the radius of convergence.

How do I calculate the coefficients of a Taylor series?

The coefficients of a Taylor series can be calculated using the Taylor series formula, which involves taking derivatives of the function at the point of expansion. Alternatively, there are also specific formulas for common functions, such as sine, cosine, and exponential functions.

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