Find the general expression of the kth nonzero term in the taylor series f(x) = 3/(1+x), for k= 0, 1, 2,...
To find the kth nonzero term in the Taylor series expansion of f(x) = 3/(1+x), we can use the formula for the Taylor series coefficients. The general expression for the kth nonzero term in the Taylor series expansion is given by:
a_k = f^(k)(a) / k!
where f^(k)(a) denotes the kth derivative of f(x) evaluated at the point a, and k! represents the factorial of k.
Let's start by finding the derivatives of f(x):
f(x) = 3/(1+x)
f'(x) = -3/(1+x)^2
f''(x) = 6/(1+x)^3
f'''(x) = -24/(1+x)^4
f''''(x) = 120/(1+x)^5
From these derivatives, we can see that f^(k)(x) can be written as:
f^(k)(x) = (-1)^(k+1) * k! * 3/(1+x)^(k+1)
Now, let's substitute a = 0 into the formula and calculate the kth nonzero term:
a_k = f^(k)(0) / k!
a_k = (-1)^(k+1) * k! * 3/(1+0)^(k+1) / k!
Simplifying the expression further, we have:
a_k = (-1)^(k+1) * 3 / (k+1)
Therefore, the general expression for the kth nonzero term in the Taylor series expansion of f(x) = 3/(1+x) is:
a_k = (-1)^(k+1) * 3 / (k+1)
For k = 0, 1, 2, ..., this expression will give you the corresponding terms in the Taylor series of f(x).