Find a power series representation for the function. (Give...

Find a power series representation for the function. (Give your power series representation centered at $x = 0$.) $g(x) = x^6 \ln(1 + x)$ $\sum_{n=1}^{\infty}$ Evaluate the indefinite integral as a power series. $\int x^6 \ln(1 + x) dx$ $f(x) = C + \sum_{n=1}^{\infty}$ What is the radius of convergence R? $R = $

Transcript

00:01 For this problem we are to evaluate as a power series the indefinite integral of arc tangent of x cubed d x.

00:08 Now we have to recall that the power series representation for arc tangent of x is equal to the summation from n equal 0 to infinity of negative 1 raise a power of n times x raise 2 n plus 1 all over 2n plus 1.

00:30 Our integral of arc tangent of x cubed dx as a power series would be the integral of the summation from n equals 0 to infinity of negative 1 raise a power of n and then instead of writing x you will replace it by x cubed so that's x cubed raise a power of 2n plus 1 all over 2n plus 1 and then we have dx.

01:03 So then this is equal to the integral of the summation from n equals 0 to infinity of negative 1 raise to n times x raised to 3 times 2n plus 1 is 6n plus 3 all over 2n plus 1 and then dx.

01:23 Expanding this, we have summation from n equals 0 to infinity of negative 1.

01:31 Raise to n times x raised to 6 n plus 3 over 2 n plus 1 becomes x cubed for the first term minus we have x raised to 9 all over 3 plus we have x raised to a power of 15 all over 5 minus x raised to 21 all over 7 that's when n is 3 and so on we'll just expand this until the fourth term because later when we get the antiderivative of this, the fifth term will be the constant c since we have to include it.

02:17 So now this is equal to x raised to a power of 4 over 4 minus x raise a power of 10 over 3 times 10 plus x raise to 16 over 5 times 15 plus 5 times 15 plus 1 that's 16 minus x raised to 21 plus 1 or 22 all over 7 times 22 and so on and then plus c simplifying that we have x raised a power of 4 over 4 minus x raised a power of 10 over 30 plus x raise to 16 over 80 minus x raise a power of 22 all over 154 and then plus and so on.

03:04 This is the same as x -rays of power of 4 over 4 plus negative x -res of power of 10 over 30 plus x -rays to 16 over 80 plus negative x -rays the power of 22 over 154 plus c and so on.

03:26 So these are the terms you have to write.

03:30 And for the radius of convergence, we need the summation form of this antiderivative, and this is just summation from n equals 0 to infinity of negative 1 raise of n times x raise to 6n plus 3 plus 1 all over 2n plus 1 times 6n plus 3 plus 1.

03:59 Remember we're only integrating with respect to x.

04:03 So the n here will be deemed constant.

04:07 And then we have plus c.

04:09 So that's the same as summation from n equals 0 to infinity of negative 1 raise of power of n times x raised to 6n plus 4 all over 2 n plus 1 times 6n plus 4...

Question

Please give Ace some feedback