Estimating a sum with an error of 0.00001 (solution included)

[s3a]

The question and solution are both attached. I am stuck at the inequality between the sum and the integral right before stating that the integral's sum is 1/(4k^4).

I am confused as to how I should compare a sum and an integral. I know Integrals are defined as Riemann sums but, to be honest, I forgot about that stuff completely.

Basically, what's the logic behind the comparison?

Any input would be greatly appreciated!
Thanks in advance!

 

[Dick]

Look at the pictures here. http://tutorial.math.lamar.edu/Classes/CalcII/IntegralTest.aspx The series in this case represents the area of the rectangles that are below the curve. So the area of the curve (which you can compute with an integral) is greater than the sum of the series. It's important here that 1/x^5 is decreasing and positive, so the rectangles lie below the curve.