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

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The discussion centers on estimating a sum with an error margin of 0.00001, specifically comparing a sum to an integral in the context of Riemann sums. The integral in question is derived from the function 1/x^5, which is decreasing and positive, indicating that the area under the curve (computed via integral) exceeds the sum of the series represented by the rectangles below the curve. The conclusion emphasizes the importance of understanding the relationship between sums and integrals, particularly in calculus.

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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!
 

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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.
 

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