Please show each step I am having trouble with power functio

Please show each step. I am having trouble with power functions and I want to see this problem done step-by-step. Thank you so much for your help.

Solution

First, generate a power series for ln(1 + x).

By the infinite geometric series:
1/(1 - x) = sum(n=0 to infinity) x^n.

Replacing x with -x gives:
1/[1 - (-x)] = sum(n=0 to infinity) (-x)^n
==> 1/(1 + x) = sum(n=0 to infinity) [(-1)^n * x^n].

Then, integrating both sides:
1/(1 + x) dx = sum(n=0 to infinity) [(-1)^n * x^n] dx
==> ln|1 + x| = sum(n=0 to infinity) [(-1)^n * x^(n + 1)]/(n + 1).

Since this converges for |x| < 1 only:
ln(1 + x) = sum(n=0 to infinity) [(-1)^n * x^(n + 1)]/(n + 1).

Finally, replacing x with x^4 gives:
ln(1 + x^4) = sum(n=0 to infinity) [(-1)^n * (x^4)^(n + 1)]/(n + 1)
==> ln(1 + x^4) = sum(n=0 to infinity) [(-1)^n * x^(4n + 4)]/(n + 1).

So, applying the FTC gives:
ln(1 + x^4) dx (from x=0 to 0.4)
= sum(n=0 to infinity) [(-1)^n * x^(4n + 4)]/(n + 1) (from x=0 to 0.4)
= sum(n=0 to infinity) [(-1)^n * x^(4n + 5)]/[(n + 1)(4n + 5)] (eval. from x=0 to 0.4)
= sum(n=0 to infinity) [(-1)^n * 4^(4n + 5)]/[10^(4n + 5) * (n + 1)(4n + 5)].

Then, by the alternating series remainder:
|R_n| <= a_n+1.

With a_n+1 = 4^(4n + 9)]/[10^(4n + 9) * (n + 2)(4n + 9)]
|R_n| <= 4^(4n + 9)]/[10^(4n + 9) * (n + 2)(4n + 9)].

Since we require 6 decimal place accuracy, we want:
4^(4n + 9)]/[10^(4n + 9) * (n + 2)(4n + 9)] <= 10^(-7)
(Note that I chose 10^(-7) to make the 7th decimal place uncertain)

The least integer solution to this n = 2. Thus:
ln(1 + x^4) dx (from x=0 to 0.4)
= sum(n=0 to infinity) [(-1)^n * 4^(4n + 5)]/[10^(4n + 5) * (n + 1)(4n + 5)]
sum(n=0 to 2) [(-1)^n * 4^(4n + 5)]/[10^(4n + 5) * (n + 1)(4n + 5)]
0.002033.