Sequences and series.

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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:

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

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

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:

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

-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: $\frac{1}{x^2 + 1}$

Write:

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

and use:

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

Regards.

Fernando Revilla
Nov 24th 2010, 12:52 PM
track02

Thanks for the assistance.

Correct me if I am wrong, but by treating [LaTeX ERROR: Convert failed] as [LaTeX ERROR: Convert failed] 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
$ a_n=n^2. $

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

$ \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 [LaTeX ERROR: Convert failed] as [LaTeX ERROR: Convert failed] 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