Use a power series to approximate the following definite integral correct to five decimal places dx/(1+x^6) from 0 to 1/2

Question

anonymous · Answer

$$\int\limits_{0}^{1/2} dx/(1+x^6)$$

anonymous · Answer

I want to make sure I did this right...

anonymous · Answer

This can be rewritten using the geometric series for |x| < 1, I think. Then, maybe do the integral test? How did you tackle this mate?

anonymous · Answer

Well, in class we did use the geometric series and we came up with 1/1-x and I substituted x=-x^6 into that equation and tried to solve it, but I am not too sure if that is the right way to do it...

anonymous · Answer

Yeah; this looks correct. It's an alternating series, so it becomes easier to approximate :-).

anonymous · Answer

Ok, so then I would just use the alternating series test on $$\sum_{0}^{1/2}(-1)^nx^(6n)$$?

anonymous · Answer

supposed to be x^6n...

anonymous · Answer

That should be to infinity :-). Just start summing terms until you are done.

dumbcow · Answer

http://www.wolframalpha.com/input/?i=integrate+dx%2F%281%2Bx^6%29+ look at series expansion i believes this is the infinite sum $$\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{6n+1}}{6n+1}$$

anonymous · Answer

okay, so the limits do not matter?

dumbcow · Answer

evaluate at x = 1/2 ......sum = 0 for x=0

anonymous · Answer

Yea, I used wolfram, but it didn't clear up my confusion

anonymous · Answer

Look at the second to last line: http://www.physicsforums.com/showthread.php?t=49916 You first get a power sum representation for the function, then evaluate the power sum in the limits.

anonymous · Answer

Ohhhhh, okay!! I see now!! Thank you all very much!

anonymous · Answer

No problem :-)