use the following information to complete parts a. and b. below.
f(x)=2/x , a = 2
What are the first four terms?
Write power series using summation notation.
The power series representation for the function f(x) = 2/x is:
f(x) = 2/x = 2(1/x) = 2(x^(-1))
To find the power series representation for f(x), we need to find the coefficients of the terms (a_n) in the power series expansion. In this case, the formula for the general term a_n is:
a_n = f^(n)(a)/n!
where f^(n)(a) represents the nth derivative of f(x) evaluated at x = a.
Using the formula, we can find the first four terms of the power series representation of f(x) centered at a = 2:
a_0 = f(2)/0! = 2/2 = 1
a_1 = f'(2)/1! = -2/2^2 = -1/2
a_2 = f''(2)/2! = 4/2^3 = 1/2
a_3 = f'''(2)/3! = -12/2^4 = -3/4
Therefore, the first four terms of the power series representation of f(x) centered at a = 2 are:
1 - (1/2)(x - 2) + (1/2)(x - 2)^2 - (3/4)(x - 2)^3
In summation notation, the power series representation can be written as:
f(x) = Σ[ n=0 to ∞ ] (a_n)(x - a)^n = 1 - (1/2)(x - 2) + (1/2)(x - 2)^2 - (3/4)(x - 2)^3 + ...