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September 30th, 2009, 07:52 AM   #1
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error term for arctan(x)

The question reads:
Find a vale of n so that any partial sum with at least n terms is within 1/100 of the target value of pi/4 of the series 1-1/3+1/5-1/7+...

From our book and notes in class this is the arctan series where you let x = -y^2. I know that the error term is employed to find n. If my notes are correct then the error terms is the intergral of (-y)^n / 1+y^2 dy. (Sorry for not knowing how to format it probably). Another student in my class said that the error term is actually the intergral of y^n dy. Which one is it? Thank you for the help.

-TomC
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October 22nd, 2009, 08:49 AM   #2
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Re: error term for arctan(x)

if

x_1 - x_2 + x_3 - x_4 + ...

is an alternating series (x_i > 0 for all i) with x_i's strictly decreasing, then you can show that the partial sum

x_1 - x_2 + x_3 - x_4 + ... + (-1)^(n+1) x_n

is within x_n of the limit. No arctan required (of course we're assuming that the problem isn't lying and the limit actually is pi/4).
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