# Taylor Series using Geometric Series and Power Series

1. Sep 29, 2010

### jegues

1. The problem statement, all variables and given/known data
See figure attached.

2. Relevant equations

3. The attempt at a solution

Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the,

$$\frac{1}{x^{2}}$$

$$\int x^{-2} = \frac{-1}{x} + C$$

How do I deal with the C?

If I can get,

$$\frac{-1}{x}$$

I can work with it to get something like the following,

$$\frac{\text{first term of geometric series}}{1 - \text{common ratio}}$$

So what do I do about the C? Once I figure this out I can make more of an attempt into shaping,

$$\frac{-1}{x}$$

into the form mentioned above.

Any ideas?

Thanks again!

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Last edited: Sep 29, 2010
2. Sep 30, 2010

### vela

Staff Emeritus
You're going about it backwards. Use

$$\frac{1}{x^2} = -\frac{d}{dx}\left(\frac{1}{x}\right)$$

3. Sep 30, 2010

### jegues

Alrighty I think I've got a series for,

$$\frac{1}{x^{2}}$$

See figure attached. Is this correct?

I can't seem to figure out how to express it sigma notation however.

Any ideas?

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4. Sep 30, 2010

### vela

Staff Emeritus
No, it looks like you integrated the series, but you want to differentiate -1/x to get 1/x2.

5. Sep 30, 2010

### jegues

Whoops!

How does this look? (See figure attached)

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6. Sep 30, 2010

### vela

Staff Emeritus
Looks good!