Calculus Integrals Radius Series

label Calculus
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Evaluate the indefinite integral as an infinite series Integral of sinx/2x Find the first five non-zero terms of series representation centered at x=0.

What is the radius of convergence? Need Step by step to understand please. Thanks

May 12th, 2015

Thank you for the opportunity to help you with your question!

We know that the Maclaurin series for sine is given by:
sin(x) = Σ(n=0 -> infinity) (-1)^n * (x ^ (2n+1))/(2n+1)!
= x-x^3/3! + x^5/5! - x^7/7! + ...

Substituting this into the expression sin(x)/(2x), we get:
sin(x)/(2x) = 1/2 - x^2/(2*3!) + x^4/(2*5!) - x^6/(2*7!) + ...

and then you can integrate this easily since it is just a polynomial. You can then write out the first 5 terms to get the series representation for f(x) = sin(x)/(3x) .

Please let me know if you need any clarification. I'm always happy to answer your questions.
Jun 16th, 2015

R is infinte

Jun 16th, 2015

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May 12th, 2015
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May 12th, 2015
Sep 24th, 2017
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