The maclaurin series for ln(1+x) is given by x - x^2/2 + x^3/3 - x^4/4 + + (-1)^(n+1) x^n/n + on its interval of convergence, this series converges to ln(1+x). let f be the function defined f(x) = x ln(1 + x/3). (a) write the first four nonzero terms and the general term of the maclaurin series for f. (b) determine the interval of convergence of the maclaurin series for f. show the work that leads to your answer. (c) let p₄(x) be the fourth-degree taylor polynomial for f about x = 0. use the alternating series error bound to find an upper bound for |p₄(2) - f(2)|.

(a)

Replacing x with x/3 and then multiplying both sides by x



The general term is



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(b)

Due to the powers present in the general term, use the ratio test



The series converges on x ∈ (-3,3)
Determining the interval of convergence, test the endpoints

For x = -3, general term becomes







Diverges at x = -3.

For x = -3, general term becomes



This is an alternate harmonic series, so it converges as the terms are decreasing in size (approaching zero as n approaches infinity)

The interval of convergence is

-3 < x ≤ 3

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(c)

We wrote all the terms above in part (a). The neglected term in the alternating series of f is



Thus