Taylor Series Problem

**defjammer91** · May 5th 2009, 03:24 PM

Do I plug in pi/2 into the general cosx series expansion? Also, what would the general term be?

The question is attached in a file.

**Danny** · May 5th 2009, 03:38 PM

Originally Posted by defjammer91

Do I plug in pi/2 into the general cosx series expansion? Also, what would the general term be?

The question is attached in a file.

No, the point of expansion is $\frac{\pi}{2}$ so we use

$<br /> f \left( \frac{\pi}{2}\right) + <br /> f '\left( \frac{\pi}{2}\right) \left(x - \frac{\pi}{2} \right) + <br /> f '' \left( \frac{\pi}{2}\right) \frac{\left(x - \frac{\pi}{2} \right)^2 }{2!} + \cdots <br />$

**skeeter** · May 5th 2009, 03:40 PM

Originally Posted by defjammer91

Do I plug in pi/2 into the general cosx series expansion? Also, what would the general term be?

the question asks for the first three non-zero terms of the taylor series for cosx centered at pi/2

$f(x) = \cos{x}$

$f(x) = f\left(\frac{\pi}{2}\right) + f'\left(\frac{\pi}{2}\right)\left(x - \frac{\pi}{2}\right) + \frac{f''\left(\frac{\pi}{2}\right)\left(x - \frac{\pi}{2}\right)^2}{2!} + ... + \frac{f^{n}\left(\frac{\pi}{2}\right)\left(x - \frac{\pi}{2}\right)^n}{n!} + ...$

note that you'll end up with an odd degree taylor series

**defjammer91** · May 5th 2009, 03:51 PM

when you do the ratio test to find the interval of convergence, do you simplify anything first, or do you just plug the general term right into the ratio test? if you do have to plug in the general term, as is, into the ratio test, how does that work out? what cancels on the top and bottom?

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