Taylor Series using Geometric Series and Power Series

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,

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

[tex]\int x^{-2} = \frac{-1}{x} + C[/tex]

How do I deal with the C?

If I can get,

[tex]\frac{-1}{x}[/tex]

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

[tex]\frac{\text{first term of geometric series}}{1 - \text{common ratio}}[/tex]

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

[tex] \frac{-1}{x}[/tex]

into the form mentioned above.

Any ideas?

Thanks again!

Attached Thumbnails

   

vela

You're going about it backwards. Use

[tex]\frac{1}{x^2} = -\frac{d}{dx}\left(\frac{1}{x}\right)[/tex]

jegues

Alrighty I think I've got a series for,

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

See figure attached. Is this correct?

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

Any ideas?

Attached Thumbnails

 

vela

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

jegues

Whoops!

How does this look? (See figure attached)

Attached Thumbnails

 

vela

Looks good!