# Sequences and series.

• Nov 24th 2010, 09:50 AM
track02
Sequences and series.
Hey there.

I'm wondering if anyone here will be able to assist with the following issues:

1. I'm currently working on questions asking me to:

- $\displaystyle Find \: \alpha \: so \: that \: a_n \: = \: O(n^\alpha)$

- A sequence is then given such as $\displaystyle n^2 \: + \: n \: + \:1$

I'm curious as to what type of questions these are, they are listed under "Order of magnitude" in my notes but searching for this returns no useful resources.

2. MacLaurin Series

-Lastly, I'm struggling in creating the MacLaurin Series for the function:

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

-I have tried differentiating but this becomes quite difficult after the third derivative or so. I am guessing there is another method to solve this?

Any help concerning either of these issues would be greatly appreciated.

Thank you.
• Nov 24th 2010, 10:06 AM
FernandoRevilla
Quote:

Originally Posted by track02
-Lastly, I'm struggling in creating the MacLaurin Series for the function: $\displaystyle \frac{1}{x^2 + 1}$

Write:

$\displaystyle f(x)=\dfrac{1}{1+x^2}=\dfrac{1}{1-(-x^2)}$

and use:

$\displaystyle \displaystyle\sum_{n=0}^{+\infty}t^n=\dfrac{1}{1-t}\quad (|t|<1)$

Regards.

Fernando Revilla
• Nov 24th 2010, 12:52 PM
track02
Thanks for the assistance.

Correct me if I am wrong, but by treating $\displaystyle x^2$ as $\displaystyle (-x^2)$ you are allowing the use of sigma notation for an infinite geometric series?

Which will have the same effect as using a MacLaurin Series?

Also, does anyone have any idea where I can find some information regarding my first issue, I've been searching and still nothing!

Thanks again.
• Nov 24th 2010, 03:04 PM
zzzoak
For large n we have
$\displaystyle a_n=n^2.$

$\displaystyle O(n^{\alpha})$
means that

$\displaystyle \displaystyle { \lim \frac{a_n}{n^{\alpha}}=const }$

that is that these functions have the same order of magnitude.
• Nov 25th 2010, 12:55 AM
FernandoRevilla
Quote:

Originally Posted by track02
Correct me if I am wrong, but by treating $\displaystyle x^2$ as $\displaystyle (-x^2)$ you are allowing the use of sigma notation for an infinite geometric series?

Right.

Quote:

Which will have the same effect as using a MacLaurin Series?
Right, as a consequence of unicity of the Taylor expansion.

Regards.

Fernando Revilla