I'll ignore the minus signs because we can make them with later.
....
Did you see the pattern?
First find .
And that makes,
Ok, so the instructions state to apply Taylor's Theorum to find the power series (centered at c) for the function, and find the radius of convergence.
Problem: with the center at
I took the first through the fourth derivative and then evaluated them at the center to get an idea of the pattern, here's what I got:
1st Derivative: Evaluation:
2nd Derivative: Evaluation:
3rd Derivative: Evaluation:
4th Derivative: Evaluation:
I can use the Taylor series formula easily enough, however, I'm not sure how they came up with the answer for which holds true when n is equal to or greater than 2.
Now, I can write up the power series and perform the ratio test to find the radius of convergence on my own easily enough, however, I'm unfamiliar with this notation. I understand the in the answer because that must be true for the signs to alternate back and forth. Due to the denominators in each evaluation, I can also see where comes into play since the denominator is just 2 raised to an exponent equal to n. What I don't see is where that frickin' comes into play. Can anyone explain to me what that notation is, why it's part of the answer and how I might have to treat it when taking a limit? Thanks in advance guys.