Maclaurin Series f(x) = (2x)/(1+x2)

[jacoleen]

Homework Statement

Write the Machlaurin series for :

f(x) = (2x)/(1+x²)

Homework Equations

The Attempt at a Solution

I tried finding all the derivatives (aka f(x), f'(x), f''(x), etc..) but the equations started getting longer and longer and would always result in 0 when x=0. This was on a final exam last year, and I don't think they would have made students do long computations like that

Also, I tried just finding the derivatives for (1-x)^-1, but again, I ended up with a bunch of zeros

Help?

 

[azure kitsune]

Hmmm... I'm not sure, but would it be cheating to use that

1 + r + r^2 + r^3 + \cdots = \frac{1}{1-r}

for |r| < 1?

If you do that, you wouldn't need any calculus at all (except for the uniqueness of Maclaurin series).

 

[jacoleen]

I was thinking of doing that in the first place, but on another exam there was a similar question and part c was Find f^2007(0) :(

 

[azure kitsune]

In that case, you can work backwards, because the coefficient of xⁿ in the series gives you \frac{f^{n}(0)}{n!}

 

[jacoleen]

so my series would just be: Summation of [(-1)ⁿ2x²ⁿ⁺¹]?

I'm not sure how I would work backwards though :|

 

[azure kitsune]

Usually, you would compute \frac{f^{2007}(0)}{2007!} to find the coefficient of x²⁰⁰⁷ in a Maclaurin series. But in this case, you know the coefficient. What does that tell you about \frac{f^{2007}(0)}{2007!} ?

 

[jacoleen]

I'm really not getting it..

do i isolate for f(2007) by equating it to the summation? (without the summation term in front)

 

[azure kitsune]

You're on the right track. What is the coefficient of x²⁰⁰⁷ in this problem?

 

[jacoleen]

the summation divided by x^-2007?

 

[jacoleen]

* without the negative

 

[azure kitsune]

Actually, you found that

\frac{2x}{1+x^2} = \sum_{n=0}^{\infty}(-1)^n2x^{2n+1}

When you write out the summation, you get

\frac{2x}{1+x^2} = 2 x-2 x^3+2 x^5-2 x^7+2 x^9-2 x^{11} + \cdots

From this, can you find the coefficient of x²⁰⁰⁷?

 

[jacoleen]

is it just -2?


(i'm so sorry I logged off btw, my computer overheat :(

 

[azure kitsune]

Yep!

Now remember that the Maclaurin series of f(x) can be found by:

f(x) = \frac{f(0)}{0!} + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + \frac{f'''(0)}{3!} x^3 +\cdots

So the coefficient of x²⁰⁰⁷ must be

\frac{f^{2007}(0)}{2007!}

But you know that the coefficient is -2. So what does this tell you about f²⁰⁰⁷(0)?

 

[jacoleen]

it's equalled to 2?..and so f(2007) = 2*2007!?

 

[azure kitsune]

Close! It's actually f⁽²⁰⁰⁷⁾(0) = -2 * 2007!, but I think you got the idea. ;)

 

[jacoleen]

OMG..it actually makes sense!

Thank you so much for your help! :D

 

[Mark44]

Homework Statement

Write the Machlaurin series for :

f(x) = (2x)/(1+x²)

Homework Equations

The Attempt at a Solution

I tried finding all the derivatives (aka f(x), f'(x), f''(x), etc..) but the equations started getting longer and longer and would always result in 0 when x=0. This was on a final exam last year, and I don't think they would have made students do long computations like that

Also, I tried just finding the derivatives for (1-x)^-1, but again, I ended up with a bunch of zeros

Help?

If all you need to do is get the Maclaurin series for 2x/(1 + x^2), there's something you can do that's much simpler than what I've seen in this thread - just polynomial long division to divide 2x by 1 + x^2. Doing this, I get 2x - 2x^3 + 2x^5 -...