A. Find the first four nonzero terms of the Maclaurin series...

a. Find the first four nonzero terms of the Maclaurin series for the given function.
b. Write the power series using summation notation.
c. Determine the interval of convergence of the series.
$$f(x)=\log _{3}(x+1)$$

Key Concepts

Maclaurin Series

A Maclaurin series is a form of Taylor series expansion around zero. It represents a function as an infinite sum of terms calculated from the function’s derivatives at zero, and it provides a way to approximate functions locally using polynomials.

Series Expansion of Logarithmic Functions

The expansion of logarithmic functions, such as using the power series for ln(1+x), translates a logarithmic function into an alternating sum of power terms. This series has a specific radius of convergence and is derived by integrating the geometric series.

Change of Base in Logarithms

The change of base formula allows one to convert logarithms from one base to another, typically transforming an expression with an arbitrary logarithm base into one involving natural logarithms. This is useful because the natural logarithm has well-known series properties.

Summation Notation

Summation notation offers a compact and systematic way to represent the sum of a series using the sigma symbol. It captures the general term, the index of summation, and the limits over which the series is summed, making complex series more manageable.

Interval of Convergence

The interval of convergence defines the set of variable values for which a power series converges to the function. Determining this interval usually involves applying tests such as the ratio or root test, ensuring that the series expansion represents the function accurately within that range.

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