Well, the accuracy, for any finite polynomial expansion, deteriorates as p gets farther from -4 but is the accuracy really relevant? You are only asked to "Find first three non zero terms".seboastien said:Homework Statement
Find first three non zero terms in series expansion where the argument of funstion is small
ln(5+p)
Homework Equations
The Attempt at a Solution
The only way I could think how to do this is by saying ln(5+p) = ln(1+(4+p)) and expanding to
(4+p)- 1/2(4+p)^2 + 1/3(4+p)^3 - ... however, I imagine that this would only work if p was approx -4.
The series expansion of the logarithmic function is a mathematical representation of the logarithmic function using a sum of terms with increasing powers of the input variable. It is often used to approximate the value of the logarithm for values outside of the range where it can be directly evaluated.
The series expansion of the logarithmic function is derived by using the Taylor series expansion, which is a method of representing a function as an infinite sum of terms. This is done by taking derivatives of the function at a specific point and evaluating them at that point.
The formula for the series expansion of the logarithmic function is: ln(x) = (x-1) - (x-1)^2/2 + (x-1)^3/3 - (x-1)^4/4 + ... + (-1)^(n+1)*(x-1)^n/n, where n is the number of terms in the series.
The accuracy of the series expansion of the logarithmic function depends on the number of terms used in the expansion. The more terms included, the more accurate the approximation will be. However, for values close to the center of expansion, the series will converge faster and be more accurate.
The series expansion of the logarithmic function has various applications in mathematics and science, such as in numerical analysis, physics, and engineering. It is also commonly used in computer programs to calculate the logarithm of a number.