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#1 2012-01-30 19:28:19

nevinsmith
Member
Registered: 2010-05-19
Posts: 39

How do you find the limits of convergence for a maclarin series?

Hi guys, say that I want to expand ln(1+x) using maclarin series,
   How do i find what limits the expansion will be valid for?

what if I use taylor series and center it around 1,2,3... etc

I know how to do the expansions and stuff but not really how to find what values of x the expansion will converge for.
please help me!

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#2 2012-01-30 20:28:12

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: How do you find the limits of convergence for a maclarin series?

Hi;

The Mclaurin series expansion of ln(1+x) is

One way of knowing the radius of convergence is by your general knowledge of series. It is easy to see the when x>1 that series will not converge. Why? Because the terms will not be getting smaller. Same thing for x<-1.

Now let's look at x = 1.

That is the well known alternating harmonic series and is known to sum to ln(2).

How about at x = -1?

The series on the RHS is the well known harmonic series and is known to diverge.

So we have for an interval of convergence

-1 < x ≤ 1

This is one way of doing it, there are others. I consider this the minimum amount of knowledge that you should know for radii of convergence problems.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2012-01-30 20:56:00

nevinsmith
Member
Registered: 2010-05-19
Posts: 39

Re: How do you find the limits of convergence for a maclarin series?

bobbym, thanks for helping me

I still don't entirely understand this,
'One way of knowing the radius of convergence is by your general knowledge of series. It is easy to see the when x>1 that series will not converge. Why, because the terms will not be getting smaller. Same thing for x<-1.'

How did you see that the series will not converge if x>1? is it like a guess and check thing?

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#4 2012-01-30 21:01:15

nevinsmith
Member
Registered: 2010-05-19
Posts: 39

Re: How do you find the limits of convergence for a maclarin series?

Is it that the absolute value of the f(x) exponentialed term must be always be less than 1?

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#5 2012-01-30 21:05:07

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: How do you find the limits of convergence for a maclarin series?

Hi;

No, the numerator is increasing by a power and the denominator is increasing by 1. Let's say you had chosen x = 2.

The absolute value of the terms would look like this.

The numerator is growing faster than the denominator which you could verify by taking the limit of (2^n)/n.

What I am trying to do is to first give you an intuitive sense ( one based on general knowledge ) of when these series converge or not. There are tests that should have been covered before they rushed you into Taylor Series.

Have you seen these tests?

http://en.wikipedia.org/wiki/Convergent_series

If you are familiar with these then we can go on to the ratio test which will do this problem in a more methodical way.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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