Posted by Heather on Wednesday, November 28, 2012 at 9:19pm.
Usually, the efficient way to compute Taylor series is by using the Taylor series of the known standard fucntions like sin(x), cos(x) etc., instead of using the definition of the Taylor series in terms of the derivatives of the function.
In this case, you can write:
tan(x) = sin(x)/cos(x)
If we put:
tan(x) = c1 x + c3 x^3 + c5 x^5 + ...
(note that because tan(x) is n odd fucntion, it can only have odd powers of x in the Taylor expansion), then we can find the ck from:
sin(x) = [c1 x + c3 x^3 + c5 x^5...] * cos(x)
if we substitute for sin(x) and cos(x) their respective series expansions and equate equal coefficients of x:
x - x^3/3! + x^5/5! - ... =
[c1 x + c3 x^3 + c5 x^5...] *
[1 - x^2/2! + x^4/4! - ...]
The coefficient of x on the r.h.s. is c1 and this has to be 1, so c1 = 1.
Equating the coefficient of x^3 on both sides gives:
-1/6 = -1/2 + c3 --->
c3 = 1/3
Equating the coefficient of x^5 on both sides gives:
1/120 = 1/24 -1/6 + c5 ---->
c5 = 2/15
So, we have:
tan(x) = x + x/3 + 2/15 x^5 + ...
Related Questions
math - A sequence has t(sub1) = 5 and t(sub2) = 15 a. if the sequence is ...
Sequences...again - Sequence T1 is obtained by adding the corresponding terms of...
College Calculus (Binomial Series) - Expand f(x) = (x+x^2)/((1-x)^3) as a power ...
Calc II - Use the comparison or limit comparison test to decide if the following...
calculus - find the sum of the series: 1. the sum from n=1 to infinity of ((-1)^...
Calculus 2 - The following function has a series of the form the sum from n=0 to...
calculus - determine whether the series is convergent if so find sum: the sum of...
calculus - Determine whether the series is convergent o divergent and say what ...
calculus - determine whether the series is convergent if so find sum it is the ...
calculus - Another problem: determine whether the series is convergent if so ...
For Further Reading
