use the definition of a taylor series to find the Taylor series (centeredat c) for the function.
f(x)= 7/x
c=1
so what is f(x) as n goes from 0 to infinity?
To find the Taylor series for the function f(x) = 7/x centered at c = 1, we can use the definition of a Taylor series. The Taylor series representation of a function f(x) centered at c is given by:
f(x) = f(c) + f'(c)(x - c)/1! + f''(c)(x - c)^2/2! + f'''(c)(x - c)^3/3! + ...
In this case, we want to find the Taylor series for f(x) = 7/x centered at c = 1. Let's start by finding the derivatives of f(x):
f'(x) = -7/x^2
f''(x) = 14/x^3
f'''(x) = -42/x^4
f''''(x) = 168/x^5
Now, we can evaluate these derivatives at x = 1 (since c = 1) to find the coefficients in the Taylor series. We have:
f(1) = 7/1 = 7
f'(1) = -7/1^2 = -7
f''(1) = 14/1^3 = 14
f'''(1) = -42/1^4 = -42
f''''(1) = 168/1^5 = 168
Substituting these values into the Taylor series formula, we get:
f(x) = 7 + (-7)(x - 1)/1! + 14(x - 1)^2/2! - 42(x - 1)^3/3! + 168(x - 1)^4/4! + ...
Simplifying further, we have:
f(x) = 7 - 7(x - 1) + 7(x - 1)^2/2 - 7(x - 1)^3/6 + 7(x - 1)^4/24 + ...
So, the Taylor series for f(x) = 7/x centered at c = 1 is given by:
f(x) = 7 - 7(x - 1) + 7(x - 1)^2/2 - 7(x - 1)^3/6 + 7(x - 1)^4/24 + ...