To find the Taylor series for the function f(x) = rac{1}{x – c} centered at c, we start by recalling the definition of the Taylor series for a function f centered at c:
f(x) = f(c) + f'(c)(x - c) + rac{f''(c)}{2!}(x - c)^2 + rac{f'''(c)}{3!}(x - c)^3 + ext{...}First, we need to compute the derivatives of f(x) at the point x = c.
Step 1: Find the function value at c
Calculate f(c):
f(c) = rac{1}{c - c} = rac{1}{0} ext{ (undefined)}.Since f(c) is undefined, we cannot directly compute the Taylor series at this point. Instead, let’s consider the Taylor series from the point x = c onwards.
Step 2: Rewrite the function
To alleviate undefined behavior, we can express f(x) using a different formulation. Rewrite the function as:
f(x) = rac{1}{x - c} = - rac{1}{c - x} = - rac{1}{c(1 - rac{x}{c})}.Step 3: Use the geometric series
The function can be represented as a geometric series:
- rac{1}{c} rac{1}{1 - rac{x}{c}} = - rac{1}{c} igg( rac{x}{c}igg)^{0} + - rac{1}{c}igg( rac{x}{c}igg)^{1} + - rac{1}{c}igg( rac{x}{c}igg)^{2} + ext{...}Step 4: The Taylor series
The series converges for |x| < |c|, leading to:
- rac{1}{c} igg(1 + rac{x}{c} + igg( rac{x}{c}igg)^{2} + rac{x^{3}}{c^{3}} + ...igg) = - rac{1}{c} - rac{x}{c^{2}} - rac{x^{2}}{c^{3}} - rac{x^{3}}{c^{4}} - ...Thus, you can express the Taylor series centered at c for the function f(x) = rac{1}{x – c} as:
f(x) = - rac{1}{c} - rac{1}{c^{2}}(x - c) - rac{1}{c^{3}}(x - c)^{2} - rac{1}{c^{4}}(x - c)^{3} - ...Conclusion
While f(c) itself was undefined, by manipulating the function and applying the geometric series approach, we’ve derived the Taylor series effectively for values x near c (but not equal to c).