- #1

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## Homework Statement

write a power series representation of the following:

[tex] \frac{x}{15x^2 +1} [/tex]

## Homework Equations

the formula

[tex] \frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n [/tex]

## The Attempt at a Solution

we can rewrite the summnd like

[tex] \frac{x}{15} \left( \frac{1}{1+\frac{x^2}{15}} \right)[/tex]

we can write the denominator from the above term as:

[tex] 1 - \left( - \left( \frac{x}{\sqrt{15}} \right)^2 \right) [/tex]

so using the above term we can write the series like:

[tex] \frac{x}{15} \sum_{n=0}^∞ (-1)^n \frac{x^{2n}}{15^{n/2}}[/tex] /known data[/b]

and this simplifies to:

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

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:

[tex] \frac{x^2}{\sqrt{15}} < 1[/tex]

[tex] x < \sqrt{\sqrt{15}} [/tex]

This means that the interval is

[tex] \left( -15^{1/4} , 15^{1/4} \right) [/tex]

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!