# Math Help - Help me! This power series is driving me crazy!

1. ## Help me! This power series is driving me crazy!

The function is represented as a power series

Find the first few coefficients in the power series.
C0-C5

I've obtained C0 and the radius of convergence but every other answer I put in keeps coming back as incorrect.
I'm going by 1/(1-x) = 1 + x + x^2 + ...
So for C0 I get 1x3=3
This is where it goes downhill... For C1 I get 3[-100x^2]^1 so C1=-300?? and I sort of repeat this method except each time the final exponent gets larger.
I found the radius of convergence by doing |100x^2| < 1 so x < 1/10
Please help with the rest! I've been doing this problem FOREVER and I don't understand why I at very least haven't guessed the answers!

2. Originally Posted by sgares
The function is represented as a power series

Find the first few coefficients in the power series.
C0-C5

I've obtained C0 and the radius of convergence but every other answer I put in keeps coming back as incorrect.
I'm going by 1/(1-x) = 1 + x + x^2 + ...
So for C0 I get 1x3=3
This is where it goes downhill... For C1 I get 3[-100x^2]^1 so C1=-300?? and I sort of repeat this method except each time the final exponent gets larger.
I found the radius of convergence by doing |100x^2| < 1 so x < 1/10
Please help with the rest! I've been doing this problem FOREVER and I don't understand why I at very least haven't guessed the answers!
$\frac{3}{1+100x^2}=3\cdot\frac{1}{1+(10x)^2}=3\cdo t\sum_{n=0}^{\infty}(-1)^n(10x)^{2n}$

use the Root test

3. use the Root test
What do you mean by "use the root test"? How will that help me find the coefficients?
Also, using your formula and simply plugging in n=1, n=2, n=3, and n=4 I received the coefficients 3, -300, 30000, -30000000, 300000000... these are the same answers I was receiving prior to this, and they were all wrong besides the first one.

4. Originally Posted by sgares
What do you mean by "use the root test"? How will that help me find the coefficients?
Also, using your formula and simply plugging in n=1, n=2, n=3, and n=4 I received the coefficients 3, -300, 30000, -30000000, 300000000... these are the same answers I was receiving prior to this, and they were all wrong besides the first one.
You asked for the radius of convergence, and that is the right power series.

5. I believe you that it is the correct power series, but why am I getting incorrect answers for the coefficients then? Is it something wrong with my calculations or with my webwork? Also, I stated that I already calculated the radius of convergence, and it was 1/10.

As I just stated, I received the coefficients 3, -300, 30000, -30000000, 300000000, and the final four of these are incorrect. Why though? :\

6. Originally Posted by sgares
I believe you that it is the correct power series, but why am I getting incorrect answers for the coefficients then? Is it something wrong with my calculations or with my webwork? Also, I stated that I already calculated the radius of convergence, and it was 1/10.

As I just stated, I received the coefficients 3, -300, 30000, -30000000, 300000000, and the final four of these are incorrect. Why though? :\
That should be right...why do you say it is wrong?

7. I don't know man... I'm so flustered at this point from this problem! Grrr!

8. It is probably because you have found the first five nonzero coefficients. I suspect the answer is
3, 0, -300, 0, 30000

-Dan

9. Woah, that was correct topsquark... thank you so much for that lol. I still have no idea why that is correct though, so if anyone could explain this to me that would make my day a lot better!

Seriously though, if I could I would sit here thanking you all day for coming up with that answer. That just made me feel so much better.

10. Originally Posted by sgares
Woah, that was correct topsquark... thank you so much for that lol. I still have no idea why that is correct though, so if anyone could explain this to me that would make my day a lot better!

Seriously though, if I could I would sit here thanking you all day for coming up with that answer. That just made me feel so much better.
Do you see how the exponent of $x^{2n}$ is going to be even since $n\in\mathbb{N}$?

Well did you wonder what happened to the even terms?

The reason is that if we let

$f(x)=\frac{1}{1+100x^2}$

and let $f^{(n)}(0)$

We see that

$f^{(n)}(0)=\left\{ \begin{array}{cc} (-1)^n\cdot{10}^{2n} & \mbox{ if } {\text{n is even}}\\ 0 & \mbox{ if } {\text{n is odd}}\end{array}\right.$

So since two (three if excluding zero) of the first terms are odd numbers, we can see that they are zero.

11. Another piece of advice. The problem asked you to find the Taylor series about x = 0. What Mathstud (and presumably you) derived was based on a geometric series summation. The two give identical answers, but if you did the actual work with the Taylor series you would have seen the coefficients that were 0 directly.

That being said I would probably have given the same answer as you as such problems typically ask for only the non-zero coefficients.

-Dan