Well what is the MacLaurin Series for sin(x)? What do you get when you truncate it to 1, 3, 5 places? What do you get when you substitute pi/10?
I still not getting how even start to answer a question related to Taylor Polynomials so please I need some strong help and advice in how to tackle such questions as this ones above:
i) Use Taylor polynomials p1,p3,p5,... about 0, successively, to evaluate sin(pi/10) to four decimal place showing all your working.
ii) Use multiplication of Taylor series to find the quartic Taylor polynomial about 0 for the function f(x) = e^x - 1 / sqr of 1+x
evaluating the coefficients
I don't need just an answer I also need I follow trough with explanations of what is happening.
xxx
Hi Prove It,
Following your instruction:
First I got the Taylor series for sin x ( why this series does not use the even terms?)
Then I started to calculate :
p1 = x - 1/3! x^3
p3 = p1 + 1/5! x^5
p5 = p3 - 1/7! x^7
p7 = p5 + 1/9! x^9
when x = pi/10
It is correct?
What about the second problem?
ii) Use multiplication of Taylor series to find the quartic Taylor polynomial about 0 for the function f(x) = e^x - 1 / sqr of 1+x
evaluating the coefficients
I'm not sure if I have done it right
First I obtained the Taylor series for e^x and the subtracted -1 from each term and ended up with the following series:
e^x - 1 = x - 1 + (1/2)x^2 - (5/6)x^3 - (23/24)x^4 - ....
Second I used the binomial series to find the taylor serie about 0 for the function 1/sqr 1+x = 1/(1+x)^1/2 = (1+x)^-1/2
1/sqr 1+x = 1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...
Third I started to multiply both series term by term:
e^x - 1/sqr 1+x = (x - 1 + (1/2)x^2 - (5/6)x^3 - (23/24)x^4 - ....) x (1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...)
=x(1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...) - 1(1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...) +((1/2)x^2)(1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...) - ((5/6)x^3)(1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...) - ((23/24)x^4)(1 - (1/2)x + (3/8)x^2 - (5/16)x^3 + (35/128)x^4 - ...) - ...
Does it makes any sense or I totally wrong?