This is a thread meant to help those who are having trouble with power series. Also I will cover some useful uses for these power series.
Part 1. What is a power series?
A power series bascially works of this observation
"Hey at x=0 and are equal, and also for numbers close to x=0 is a good approximation for . I wonder if I can find a way to find a good approximation for numbers like x=7?"
Well you do this by first taking one of these cases and saying since . Next you can say "well if they increase at a similar rate( the approximation will be better...and hey if the rate of the rate( ) increases similarly then the approximation will be even better..and if the rate of the rate of the rate....ad infinitum"
This is how The classic arises....
This series will ensure that for all the values at which the series converges the sereis will follow the pattern of the series's rate being comparable to the rate of the function being approximated.
Now I am writing this assuming you knew this. So That is all I am going to go into regarding that
Part 2. Why power series converge only for a certain values?
To show this make a show of good faith and assume that the following is correct
I will show later how this was gotten...You may ask why there has to be a limiting interval?
This is called the interval of convergence. This is exactly what it sounds like, the series only converges for x values that are an element of the interval. To show this I will take one of the values contained x=-1.
If we imput that into our series we have
Now since and we see this converges by the alternating series test
So we see that this converges for x=-1
Next let's show why x=1 is excluded. When x=1 the series becomes
which diverges by the integral test
Now lets test another random value...say Gelfond's constant
Imputting that we get
Which obviously diverges due to the n-th term test
This is also true
Part 3. How do I find power series for unfamilar functions?(I will do this for Maclaurin only)
First off let me be blunt. You are going to have much difficulty in this topic if you don't memorize the basic power series
To learn them look here Maclaurin Series -- from Wolfram MathWorld
The must knows are
From these you can use tricks to find all the others you'll most likely encounter
First lets look at the previously stated power series for
Here is how this was derived since
and we know that
we can make the jump that
Note that although integrating a power series does not change its radius of convergence it can change its endpoint behavior
I will go over three more
THe first is this: Find the power series for
Well first we need to find the power series for before we can tackle this...first we make this observation
You can verify the details missed yourself
The second one is
The key for these type is to get it into one of our normal forms
To do this we first need to have a 1-u(x) on the bottom so we rewrite it as this
The second part of which looks like our known formula so we apply the power series replacing
Check interval of convergence by using ratio test
I will be back later for third example and uses.