Thread: Power series representatin of a function

1. Power series representatin of a function

Question:

Find a power series representation for the function.

$\displaystyle f(x) = \frac{3}{1-x^2}$

My solution:

Since, $\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + .... = \sum_{n=0}^{\infty} x^n$

Then,

$\displaystyle \frac{3}{1-x^2} = \sum_{n=0}^{\infty} 3x^{2n}$

Am I correct?

2. Originally Posted by calc101
Question:

Find a power series representation for the function.

$\displaystyle f(x) = \frac{3}{1-x^2}$

My solution:

Since, $\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + .... = \sum_{n=0}^{\infty} x^n$

Then,

$\displaystyle \frac{3}{1-x^2} = \sum_{n=0}^{\infty} 3x^{2n}$

Am I correct?
looks fine.

3. And I calculated the radius of convergence to be: 1, with the following interval: (-1,1). Correct?

4. Originally Posted by calc101
Question:

Find a power series representation for the function.

$\displaystyle f(x) = \frac{3}{1-x^2}$

My solution:

Since, $\displaystyle \frac{1}{1-x} = 1 + x + x^2 + x^3 + .... = \sum_{n=0}^{\infty} x^n$

Then,

$\displaystyle \frac{3}{1-x^2} = \sum_{n=0}^{\infty} 3x^{2n}$

Am I correct?
Sorry!

5. Originally Posted by DeMath
A small clarification: for $\displaystyle x \in \left( { - \infty ;{\text{ }} - 1} \right) \cup \left( { - 1;{\text{ }}1} \right) \cup \left( {1;{\text{ }}\infty } \right)$.
Are you saying for example that $\displaystyle \frac{3}{1-2^2}=3\sum_{j=1}^\infty2^{2j}$?

6. Originally Posted by calc101
And I calculated the radius of convergence to be: 1, with the following interval: (-1,1). Correct?
Yes this correct.

Whereas
Originally Posted by DeMath
A small clarification: for $\displaystyle x \in \left( { - \infty ;{\text{ }} - 1} \right) \cup \left( { - 1;{\text{ }}1} \right) \cup \left( {1;{\text{ }}\infty } \right)$.
Is grossly misguided!