Answer the following questions about the function f(x) =...

Answer the following questions about the function f(x) = ln(1 + x^3). Part 1 - 1 point Find the Maclaurin series of f(x) = ln(1 + x^3). The Maclaurin series of f(x) is sum_{n=1}^{infty} Part 2 - 1 point Find the radius of convergence of the Maclaurin series you found in part one. R=

Transcript

00:01 For the first part of this problem, we are to find a mclaurian series for the natural log of 1 plus x cubed.

00:07 Now we recall the mclaurian series for 1 over 1 minus x is equal to the summation from n equals 0 to infinity of x raise of power of n.

00:18 So 1 over 1 plus x, that's equal to the summation from n equals 0 to infinity of negative x raise of power of n.

00:29 That's summation from n equal 0 to infinity of negative 1 raise to n times x raise a power of n.

00:36 Taking the n to derivative both sides, you have integral of 1 over 1 plus x dx, that's equal to the integral of the summation from n equal 0 to infinity of negative 1 raise to n times x rase to n dx.

00:51 That's natural log of the absolute value of 1 plus x equal to integral of 1 minus x plus x squared minus x cubed plus x 3rd with the 4th power and so on d x integrating term by term we have natural log of can just get rid of the absolute value anyway since you just want natural log of 1 plus x cubed so natural log of 1 plus x, that's equal to x minus x squared all over 2 plus x cubed over 3 minus x -rays to the 4th power over 4 and so on, and then plus c.

01:42 Note that if x is 0, you have natural log of 1 equal to 0 plus c, meaning c is also 0.

01:53 So natural log of 1 plus x, that's equal to summation from n equals 1 to infinity of negative 1 raise of power of n plus 1 times x raise the power of n all over n.

02:11 So if x is x cubed, we have natural log of 1 plus x cubed.

02:17 That's equal to summation from n equals 1 to infinity of negative 1.

02:23 Raised to n plus 1 times x raised to the third power times n or that's just 3n all over n.

02:33 For the radius of convergence we have to apply ratio test.

02:38 Now our nth term here is equal to negative 1 raise to n plus 1 times x raised to 3n all over n...

Question

Answer the following questions about the function f(x) = ln(1 + x^3). Part 1 - 1 point Find the Maclaurin series of f(x) = ln(1 + x^3). The Maclaurin series of f(x) is sum_{n=1}^{infty} Part 2 - 1 point Find the radius of convergence of the Maclaurin series you found in part one. R=

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