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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

2005 • Question 6

Question

Math 6210 • Real Analysis 2005 • Question 6

Provide a rigorous computation of the sum of the following series: \[ \sum _{n=1}^{\infty }\frac {1}{n^4}. \] (Hint: one possible approach is to consider the integral of \(f(z)=\frac {\pi }{z^4}\cot (\pi z)\) over an appropriate sequence of contours.)

Raw LaTeX

Provide a rigorous computation of the sum of the following series:
\[
\sum_{n=1}^{\infty}\frac{1}{n^4}.
\]
(Hint: one possible approach is to consider the integral of
\(f(z)=\frac{\pi}{z^4}\cot(\pi z)\) over an appropriate sequence of contours.)

Topic Tags

Cauchy Integral Theory and Morera's Theorem Complex Integration and Real Integral Evaluation Lebesgue Integration Lp Spaces and Sequence Spaces