Thread: Why do we need to represent functions as power series?

As you may have guessed, I am currently studying cal II, the section about: sequences, series, power series, et al...

I do not understand the need for representing functions as power series? I know how to mechanically find the answer (following procedure basically) but I lack the understanding behind it.

Can someone please explain it to me with some clear examples.

So what if I knew that $\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$?

Is that saying:

When x = 3 then:

$\displaystyle f(3) = \frac{1}{-2} = \sum_{n=0}^{\infty}3^n$


I am lost....

Well, maybe the above is a bad example because they are equal iff -1<x<1.

Hmm...maybe I am some what understanding it....

Have you ever tried to find this integral $\displaystyle \int_0^{0.5} {e^{ - x^2 } dx} $?

It is truly simple using series.

ever wonder how a calculator calculates the a trig function for a specific angle?

Originally Posted by Plato

Have you ever tried to find this integral $\displaystyle \int_0^{0.5} {e^{ - x^2 } dx} $?

It is truly simple using series.

Well, the method that I would use is of course integration by substitution, which will not work because du = -2xdx.

May I ask how this is simplified using series? Can you please show me?

Originally Posted by skeeter

ever wonder how a calculator calculates the a trig function for a specific angle?

Not really, but now that you mention it, I am intrigued. Of course you are suggesting that power series are being used. How? I have no idea. Again, if you care to provide an example, a really simple one, I will appreciate it!

Thanks,

Originally Posted by calc101

Not really, but now that you mention it, I am intrigued. Of course you are suggesting that power series are being used. How? I have no idea. Again, if you care to provide an example, a really simple one, I will appreciate it.

You can find in Stewart’s textbook.

Originally Posted by calc101

As you may have guessed, I am currently studying cal II, the section about: sequences, series, power series, et al...

I do not understand the need for representing functions as power series? I know how to mechanically find the answer (following procedure basically) but I lack the understanding behind it.

Can someone please explain it to me with some clear examples.

So what if I knew that $\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$?

No "what if" about it. You don't know that- it's not true! What is true is that $\displaystyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ provided that -1< x< 1

Is that saying:

When x = 3 then:

$\displaystyle f(3) = \frac{1}{-2} = \sum_{n=0}^{\infty}3^n$


I am lost....

That's a very good reason for learning about power series- so you won't make foolish mistakes like that!

Another reason for writing functions in terms of power series is to be able to extend the functions to other "number systems".

For example, we know how to add, subtract, multiply and divide complex number and matrices but what is $\displaystyle e^z$ or cos(z) or sin(z) if z is a complex number or a matrix?

Answer: write those functions as power series and use the properties of sums and products. That's why analytic functions are so important in Complex Analysis- they are the functions that can be written as power series!

It's unfortunate that most books (and instructors) introduce series and all the tests and then afterwards the applications! I wonder why not in reverse? It sure would eliminate the question why are we doing this.

Originally Posted by skeeter

ever wonder how a calculator calculates the a trig function for a specific angle?

A popular misconception but it is not usual for a calculator to use a power series to evaluate trig functions (or if a series is involved it is a very minor part of the algorithm, and then it is more likely to be a polynomial approximation than a power series if you see the distinction).

CB