How can i find a power series for this integral?

Thread starter sportlover36
Start date Apr 25, 2010
Tags

Integral Power Power series Series

In summary, the coefficients of a power series for a given integral can be determined by using a formula involving the nth derivative of the function, the center of the power series, and the value of the function at the center. The process for finding a power series for a complicated integral involves simplifying the integrand and finding coefficients using the same formula. A power series can be used to approximate the value of an integral by taking a finite number of terms, and the interval of convergence for a power series can be determined using tests. The power series is considered the best approximation for a given integral if it converges to the exact value within the interval of convergence.

Apr 25, 2010

sportlover36

how can i find a power series for this integral? [tex]\int cos(x^3)[/tex]

Physics news on Phys.org

Apr 25, 2010

tiny-tim

Science Advisor

Homework Helper

Hi sportlover36!

(have an integral: ∫ and a sigma: ∑ and try using the X² tag just above the Reply box

)

Just write out the series for cosx, and then write x³ everywhere instead of x.

Related to How can i find a power series for this integral?

1. How do I determine the coefficients of a power series for a given integral?

The coefficients of a power series can be determined by using the formula for the nth term of a power series:
c_n = 1/n! * f⁽ⁿ⁾(a) * (x-a)ⁿ
where f(x) is the function being integrated, a is the center of the power series, and f⁽ⁿ⁾(a) is the nth derivative of f(x) evaluated at a.

2. What is the process for finding a power series for a complicated integral?

The process for finding a power series for a complicated integral involves first determining the center of the power series and then finding the coefficients using the formula mentioned in the previous question. This may require simplifying the integrand and taking multiple derivatives to find the necessary coefficients.

3. Can I use a power series to approximate the value of an integral?

Yes, a power series can be used to approximate the value of an integral by taking a finite number of terms in the series. The more terms that are included, the more accurate the approximation will be.

4. Is there a specific range of values for which a power series will converge?

Yes, a power series will only converge within a certain interval, known as the interval of convergence. This interval can be determined by using the ratio test or the root test on the series.

5. How do I know if a power series is the best approximation for a given integral?

The power series is considered the best approximation for a given integral if it converges to the exact value of the integral within the interval of convergence. This can be checked by comparing the power series approximation to the actual value of the integral or by using other methods such as the Taylor series remainder theorem.