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 non-zero term in the Taylor series of f(x) = 3/(1+x), we need to expand the function into a power series and then identify the kth non-zero term.
The Taylor series expansion of f(x) is given by:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + f'''(a)(x-a)^3/3! + ...
To determine the general expression for the kth non-zero term, we need to identify the pattern in the expansion.
First, let's find the derivatives of f(x):
f'(x) = -3/(1+x)^2
f''(x) = 6/(1+x)^3
f'''(x) = -18/(1+x)^4
f''''(x) = 72/(1+x)^5
...
To find the general expression for the kth non-zero term, we substitute the derivatives and simplify the expression:
f(k)(x) = (-1)^(k-1) * k! * 3/(1+x)^k
In this expression, k! (k factorial) represents the product of all positive integers up to k.
Therefore, the general expression for the kth non-zero term in the Taylor series of f(x) = 3/(1+x) is:
(-1)^(k-1) * k! * 3/(1+x)^k