2 questions on Maclaurin series

Hello! I'm stuck on these questions... Can someone help? (Rofl)

First: how would you write f(x) = x / (1 + x^{3}) as a series?

I mean like

∞

Σ = .....

k=0

I know that 1/(1-x) =

∞

Σ x^{k}

k=0

But how do i convert this into f(x)?

Second: how would you find the radius of convergence of the following binomial series:

∞

Σ (3k k) x^{2k+1}

k=0

with (3k k) is like (n k) = n! / k!(n-k)!

I know how to do it when there's x^{k }in the sommation but with x^{2k+1}?

Thanks for the help, I've been spending hours on these two questions! (Wondering)

Re: 2 questions on Maclaurin series

Quote:

Originally Posted by

**jones123** Hello! I'm stuck on these questions... Can someone help? (Rofl)

First: how would you write f(x) = x / (1 + x^{3}) as a series?

I mean like

∞

Σ = .....

k=0

I know that 1/(1-x) =

∞

Σ x^{k}

k=0

But how do i convert this into f(x)?

Second: how would you find the radius of convergence of the following binomial series:

∞

Σ (3k k) x^{2k+1}

k=0

with (3k k) is like (n k) = n! / k!(n-k)!

I know how to do it when there's x^{k }in the sommation but with x^{2k+1}?

Thanks for the help, I've been spending hours on these two questions! (Wondering)

1. You could write , which you can then write as a geometric series.

2. Use the Ratio Test.

Re: 2 questions on Maclaurin series

Hmm I don't quite understand the second one. What difference makes the x^k in stead of x^(2k+1) in the ratio test ?

The radius of convergence of the sum with x^k should be 4/27, the sum of x^(2k+1): 2/sqrt(27)

Re: 2 questions on Maclaurin series

You have

To use the "ratio test" to determine whether the series converges, look at

Here, so

and so

Can you complete it from there?