Question

use the power series to estimate the series:

from 0 to 4 of ln(1+x)dx with absolute value of the error less than .0001/ Give your estimate of the integral as well as a bound on the error.

I found the 'terms' in the series to be:
x-(1/2)x^2+(1/3)x^3-(1/4)x^4......
with a radius of convergence of 1

I found when I plugged in 0, I got 0
how do I find the error less than .0001?

Answer 1

You need to insert te upper limit in the series, the lower limit and then subtract. However, in this that doesn't work as the series doesn't converge at the upper limit.

What you need to do is to write down the series that converges for x larger than one and the series that converges for x smaller than one, integrate both and then add up the integral of the former from x = 1 to x = 4 and the integral of the latter from x = 0 to x = 1.