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$Old$ December 4th, 2008, 01:31 PM

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I can't figure out part d. All the other parts are right.

Evaluate the integral

.
Your answer should be in the form

, where

is an integer. What is the value of

?
Hint:

(b) Now, lets evaluate the same integral using power series. First, find the power series for the function

. Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
What are the first few terms of S ?

(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by

(the answer to (a)), you have found an estimate for the value of

in terms of an infinite series. Approximate the value of

by the first 5 terms.
.
(d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.)
.

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$Old$ December 4th, 2008, 02:29 PM

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Originally Posted by amiv4 $View Post$

I can't figure out part d. All the other parts are right.

Evaluate the integral

.
Your answer should be in the form

, where

is an integer. What is the value of

?
Hint:

(b) Now, lets evaluate the same integral using power series. First, find the power series for the function

. Then, integrate it from 0 to 2, and call it S. S should be an infinite series.
What are the first few terms of S ?

(c) The answers to part (a) and (b) are equal (why?). Hence, if you divide your infinite series from (b) by

(the answer to (a)), you have found an estimate for the value of

in terms of an infinite series. Approximate the value of

by the first 5 terms.
.
(d) What is the upper bound for your error of your estimate if you use the first 12 terms? (Use the alternating series estimation.)
.

$\int_0^2\frac{dx}{x^2+4}=\frac{1}{2}\arctan\left(\frac{x}{2}\right)\bigg|_0^2=\frac{\pi}{8}$

But also note that (see other post) $\begin{aligned}\int\frac{dx}{x^2+4}=\frac{1}{4}\int\varphi\left(\frac{x^2}{4}\right)dx\end{aligned}$

See if you can get it from that

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