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

nevinsmith · 2012-01-30 19:28:19

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!

bobbym · 2012-01-30 20:28:12

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.

nevinsmith · 2012-01-30 20:56:00

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?

nevinsmith · 2012-01-30 21:01:15

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

bobbym · 2012-01-30 21:05:07

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.

Math Is Fun Forum

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

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

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

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

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

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

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

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

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

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

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