Two Power Series questions that need to be solved ly

[reichiru]

Two Power Series questions that need to be solved urgently

Homework Statement

Question 1: The function f(x) = 2x (ln(1+x)) is represented as a power series. Find coefficients c2 through c6 of the power series.
Question 2: Write a partial sum for the power series which represents the function f(x) = 1/(1+(3^2)(x^2)) consisting of the first five non-zero terms. Also find the radius of convergence.

Homework Equations

The Attempt at a Solution

Question 1: Okay so the power series I came up with is (2(-1)ⁿ)/n x (xⁿ), using the fact that ln(1+x) = 1/1+x = 1/1-(-x) and finding the partial sum 1-x+x^2-x^3... etc. So the coefficients I got were 1, 2/3, 1/2, 2/5, and 1/3. Unfortunately, the online thing says it's wrong, and I have no idea why...

Question 2: This one completely stumped me, the best I could come up with is to somehow take out the 3^2 so that you're left with 1+x^2 in the denominator, which is much easier to calculate a partial sum out of.

EDIT: Question 2 solved... just Question 1 now...
EDIT2: Never mind now, solved Question 1 as well.

 

[mighty2000]

Hello there,

For Question 1, it seems like you have the right idea but your power series is not quite accurate. The correct power series representation for f(x) = 2x (ln(1+x)) is 2x^2 + 2x^3 - 2x^4 + 2x^5 - 2x^6 + ... This can be obtained by using the power series representation for ln(1+x) which is x - x^2/2 + x^3/3 - x^4/4 + x^5/5 - x^6/6 + ... and then multiplying it by 2x. So the coefficients you should get are 0, 0, 2, 2, -2, 2, -2, ...

For Question 2, you can first write f(x) as 1/(1+9x^2) and then use the geometric series formula to find the power series representation. The first five non-zero terms are 1 - 9x^2 + 81x^4 - 729x^6 + 6561x^8. The radius of convergence can be found by using the ratio test, which in this case will give you a radius of convergence of 1/3.

Hope this helps! Let me know if you have any further questions. Good luck!