Power series representation

Homework Statement

write a power series representation of the following:
$$\frac{x}{15x^2 +1}$$

Homework Equations

the formula
$$\frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n$$

The Attempt at a Solution

we can rewrite the summnd like
$$\frac{x}{15} \left( \frac{1}{1+\frac{x^2}{15}} \right)$$
we can write the denominator from the above term as:

$$1 - \left( - \left( \frac{x}{\sqrt{15}} \right)^2 \right)$$

so using the above term we can write the series like:
$$\frac{x}{15} \sum_{n=0}^∞ (-1)^n \frac{x^{2n}}{15^{n/2}}$$ /known data[/b]

and this simplifies to:

$$\sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{15^{n/2 + 1}}$$

is that correct? This is the basis for the second part which asks for the interval of convergence
I cant write absolute value, but here goes:
$$\frac{x^2}{\sqrt{15}} < 1$$

$$x < \sqrt{\sqrt{15}}$$

This means that the interval is

$$\left( -15^{1/4} , 15^{1/4} \right)$$
Unfortunately I m getting the answer wrong as per the computer... can you please take a look and see if this is correct or not?

Thank you for your help. It is greatly appreciated!

vela
Staff Emeritus
Homework Helper

Homework Statement

write a power series representation of the following:
$$\frac{x}{15x^2 +1}$$

Homework Equations

the formula
$$\frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n$$

The Attempt at a Solution

we can rewrite the summnd like
$$\frac{x}{15} \left( \frac{1}{1+\frac{x^2}{15}} \right)$$
This isn't correct. Try multiplying out the denominator to see this.
we can write the denominator from the above term as:

$$1 - \left( - \left( \frac{x}{\sqrt{15}} \right)^2 \right)$$

so using the above term we can write the series like:
$$\frac{x}{15} \sum_{n=0}^∞ (-1)^n \frac{x^{2n}}{15^{n/2}}$$ /known data[/b]

and this simplifies to:

$$\sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{15^{n/2 + 1}}$$

is that correct? This is the basis for the second part which asks for the interval of convergence
I cant write absolute value, but here goes:
$$\frac{x^2}{\sqrt{15}} < 1$$

$$x < \sqrt{\sqrt{15}}$$

This means that the interval is

$$\left( -15^{1/4} , 15^{1/4} \right)$$
Unfortunately I m getting the answer wrong as per the computer... can you please take a look and see if this is correct or not?

Thank you for your help. It is greatly appreciated!

This isn't correct. Try multiplying out the denominator to see this.

the term should go

$$x \frac{1}{1-(-15x^2)}$$
which is
$$x \sum (-15x^2)^n$$
$$\sum (-1)^n 15^n x^{2n+1}$$

is that correct?
And it follows that:
$$\left| -15x^2 \right| < 1$$
and solving this we get
$$\left| x^2 \right| < \frac{1}{\sqrt{15}}$$

Is this correct?

Hey can you let me know if this is corrct what I did?