We all know that by partial fractions followed by telescoping sum. But if we raise the expression to the power of any positive number greater than 1, something interesting happens!
By partial fractions, .
With the knowledge of the Basel function: .
Because is the sum of all positive terms, then it must be strictly positive, then , equivalently . Thus we just found an upper bound of . Wonderful isn't it?
You could keep increasing the value of for the expression for positive number to increase the accuracy of the bound!
Try it yourself!
Question 1: Prove the partial fraction . And determine the value of . Thus show that .
Question 2: Given that and find the partial fraction of , prove that . Consider the example given and the answer you've found, can you determine a systematic way to increase the upper bound of ?
Question 3: Suppose we restrict (as mentioned above) to an odd number, why does the accuracy of the upper bound increases when the number increases?
Question 4: Consider , prove that . Similarly, consider , prove that .
Question 5 Using the answers you got from Question 2, find the partial fractions for , and thus prove the inequality
Question 6 (warning: tedious): Like in Question 4, consider , prove that .
Question 7: From Question 4 and 6, can you spot a pattern to find a lower bound of ? Prove that using this technique you've found, you can always find a rational number such that .
Question 8 (warning: very tedious): If we are further given that , and find the partial fraction of and by Cardano's method, prove that .
Question 9 (warning: extremely tedious): From Question 8, if we are again further given that , using the approaches we had before, prove that the equation below has a positive real root such that is a significantly small positive number.