Assume that (Exercises 65 and 66) and that the terms of this series may be rearranged without changing the value of the series. Determine the sum of the reciprocals of the squares of the odd positive integers.

Data from Exercise 65

The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined byWhen x is a real number, the zeta function becomes a p-series. For even positive integers p, the value ofis known exactly. For example,

Use the estimation techniques described in the text to approximate (whose values are not known exactly) with a remainder less than 10^-3.

Data from Exercise 66

In 1734, Leonhard Euler informally proved that An elegant proof is outlined here that uses the inequality

cot² x 2 2 x (provided that 0

Transcribed Image Text:

# 1 k=1k² ㅠ 2 6 .xy k³ k=1 3 = (x)₂ (d) ₂ IM* k² || k= k4 = 90' and = 1 ㅠ k6 || 6 945 (3) and (5) # XX k=1 k² T 2 6 n Σ cot² ke k=1 = n(2n 1) 3 9 for n = 1, 2, 3, . . . , where 0 = ㅠ 2n + 1