Euler found the following sums of the reciprocals of the first two even powers of the whole numbers (as well as many others). 1 = 1+ 1 1 1 + 42 1 + k2 + + .. 22 32 1 1+ 24 1 1 + +.. 44 1 + k4 k4 34 90...

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Extracted text: Euler found the following sums of the reciprocals of the first two even powers of the whole numbers (as well as many others). 1 = 1+ 1 1 1 + 42 1 + k2 + + .. 22 32 1 1+ 24 1 1 + +.. 44 1 + k4 k4 34 90 Based on Euler's work it is reasonable to guess that the sum of the cubes of the reciprocals of the whole numbers might have the same form. Consider the following conjecture. 1 = 1+ 23 1 1 1 +...+ 43 1 +...= k3 where M is a whole number, and 6 < m="">< 90.="" m="" conjecture:="" k3="" 33="" k="1" 1.="" in="" the="" previous="" written="" work="" you="" used="" the="" integral="" test="" to="" prove="" that="" the="" sum="" of="" the="" reciprocals="" of="" the="" cubes="" of="" the="" whole="" numbers,="" s="E," s="" is="" bounded="" as="" follows:="" 1="" 1="" 3="" k3="" k="1" 2="" use="" this="" result="" to="" find="" a="" more="" restrictive="" upper="" and="" lower="" bound="" for="" m="" in="" the="" conjecture.="" 8wi="">