Ehm, no, that's not the expression for the n-th coefficient of a Taylor series. (which should be in your book).beanryu said:I think you just keep taking the derivative of the previous and set x = 0
The Maclaurin Series Expansion of 5ln(7-x) is a representation of the function 5ln(7-x) as an infinite sum of powers of x, centered at x=0. It is also known as the Taylor Series Expansion at x=0.
The formula for the Maclaurin Series Expansion of 5ln(7-x) is:
f(x) = 5ln(7) + (-5/7)(x-7) + (25/98)(x-7)^2 + (-125/686)(x-7)^3 + ...
The interval of convergence for the Maclaurin Series Expansion of 5ln(7-x) is (-7, 7]. This means that the series will converge for all values of x within this interval, but may diverge for values outside of this interval.
The Maclaurin Series Expansion of 5ln(7-x) is derived by using the Taylor Series formula for a function centered at x=0. This involves finding the derivatives of the function at x=0 and substituting them into the formula. The general form for the Maclaurin Series Expansion is:
f(x) = f(0) + f'(0)x + (f''(0)/2!)x^2 + (f'''(0)/3!)x^3 + ...
The Maclaurin Series Expansion of 5ln(7-x) is useful because it allows us to approximate the value of the function for values of x within its interval of convergence. This can be helpful in situations where it is difficult or impossible to find the exact value of the function. Additionally, the Maclaurin Series Expansion can be used to find the derivatives of the function at x=0.