Find the general expression of the kth nonzero term in the taylor series for f(x)=3/(1+x) (exclude any zero terms in the series when finding this general expression)
To find the general expression for the kth nonzero term in the Taylor series of f(x) = 3/(1+x), we need to find the coefficient of the x^k term.
The Taylor series representation of a function f(x) centered at a is given by:
f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
In our case, we want to find the coefficient of the x^k term, so let's start by finding the first few derivatives of f(x):
f(x) = 3/(1+x)
f'(x) = -3/(1+x)^2
f''(x) = 6/(1+x)^3
f'''(x) = -18/(1+x)^4
Now, let's evaluate these derivatives at x = 0 to find the general expression for the kth nonzero term:
f(0) = 3
f'(0) = -3
f''(0) = 6
f'''(0) = -18
Notice that the kth nonzero term in the series will have a coefficient of (-1)^(k-1) * k!.
Therefore, the general expression for the kth nonzero term in the Taylor series of f(x) = 3/(1+x), excluding any zero terms, is:
((-1)^(k-1) * k!) * (x/a)^k
where a = 0, since we are centered at x = 0.
So, the general expression for the kth nonzero term is:
(-1)^(k-1) * k! * x^k