Power Series Representation for x/(15x^2+1): Is My Solution Correct?

[stunner5000pt]

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 can't 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]

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 can't 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!

 

[stunner5000pt]

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

You're right... my bad

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?

 

[stunner5000pt]

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

Thank you for your help

 

[vela]

Looks good except for the typo in your last line. It should be |x| not |x²|.