Now for the third, fourth, and fifth example of finding a power series( I decided to expand a little =D)

3. Find the power series for

This one is one that through experience with recognition becomes easy. The first thing you should see is the contatined within

Next comes the trick if you look closely you will see that

So now calling

we have that

and

Since I did not ask to have the interval of convergence checked we won't but to test if you are right take an element of the interval of convergence and test it in the actual function versus the power series. To pick what value to test we must realize that the power series for

is ,

so in realizing this we can say that since differentiating only effects endpoint behavior this series we derived converges AT LEAST on

So we test

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The fourth example will be this

4. Find the power series for

First we differentiate

Seeing the which by replacing

we can get the power series

So using this we see that

Therefore using this we make this jump

To test this we pick thevalue

Alternatively this could have been done by doing this

and finding the power series individually

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The fifth one is find the power series for

This can be done by recognizing

To test let

I will be back later to show some applications