Series convergent or divergent?
Sigma(n=1 to infinity) 1/(n^2-4n+5)
This is in the integral test + p-series section of our book, so please don't use the comparison test. This function is positive, continuous, and decreasing, therefore allowing the integral test, but I don't know how to solve this integral. If there is some way that I am missing, any help would be appreciated.
Update:This function is not 1/(n+1)(n-5) because that would make it 1/(n^2-4n-5), but the function is 1/(n^2-4n+5) so partial fractions cannot apply.
Comments
Complete the square in the denominator:
x² - 4x + 5 = x² - 4x + 4 + 1 = (x-2)² + 1
So
∫ dx/(x² - 4x + 5) = ∫ dx/[(x-2)² + 1] = arctan(x-2)
where the integral has lower limit 1 and upper limit ∞.
The integral is bounded, so the sum converges