For the binomial series of
If we let in the binomial series,
, if |x|<1.
The derivative of
1. Find the Taylor series for f(x) = cos (3x) centered at a = (Pi/2). Find its radius of convergence.
2. Let f(x) = (1+x)^(-1/2)
a) Find the Maclaurin series for f.
b) Use the series found in 2a to find the maclaurin series for (1+x^2)^(-1/2) and use that result to find a Maclaurin series for sinh^-1 (x).
I cannot figure out how to do these. For 1, I substituted the 3x for x in the cos(x) series... what do I do from there? I know I have to put (x-a) at the end for the center.
For 2, I have no idea what to do. It says to use the binomial series, which we didn't really cover. It's a bonus on a homework.
Can someone help with #1? I did a table of derivatives and evaluated them at Pi/2:
derivative, evaluated at Pi/2
f(x) = cos3x ,0
f '(x) = -3sin3x ,3
f ''(x) = -9cos3x , 0
f '''(x) = 27sin3x ,-27
f ''''(x) = 81cos3x ,0
f^5 (x) = -243sin3x ,243
The nth derivative would be (-1)^n * [3^(2n)], right? SO then I just used the nth derivative evaluated at (pi/2) / n! formula to get c sub n.
Summation n=0 to infinity [(nth derivative evaluated at Pi/2) / n!] * (x-Pi/2)^n
Then I substituted the actual nth derivative values to get:
0 + ([3(x-Pi/2)] / 1!) - ([27(x-Pi/2)^3] / 3!) + 0 + ([243(x-Pi/2)^5] / 5!) +...
So then I got the Taylor series of cos(3x) to be:
(-1)^n * 3^(2n+1) * (x-Pi/2)^(2n+1)
I'm really confused with this stuff. It's been awhile since I've done it and it's final time.