Problem 1694
Use residues to calculate \(\displaystyle \displaystyle\sum_{k=1}^\infty\frac1{k^2-\frac14}\). Check your result using elementary methods. Difficulty: 5. |
Problem 1696
\(\displaystyle \sum_{k=0}^\infty \frac1{k^2+k+1} = ? \) (The result should not contain any complex number!) Difficulty: 5. |
Problem 1697
Use residue calculus of the function \(\displaystyle \dfrac{\pi\ctg(\pi z)}{z^2}\) to prove that \(\displaystyle \displaystyle\sum_{k=1}^\infty\frac1{k^2}=\frac{\pi^2}6\). Difficulty: 5. |
Problem 1698
\(\displaystyle \qquad\sum_{k=1}^\infty\frac1{k^4}=? \qquad \qquad\sum_{k=1}^\infty\frac1{k^2-\frac14}=? \qquad \qquad\sum_{k=1}^\infty\frac1{k^2+1}=? \) Difficulty: 5. |
Problem 1699
\(\displaystyle \sum_{k=-\infty}^\infty\frac1{2k^2-1}=?\) Difficulty: 5. |
Problem 1700
Let \(\displaystyle N_k\) be the square with vertices \(\displaystyle \pm(k+\frac12)\pm(k+\frac12)i\). What is \(\displaystyle \frac1{2\pi i}\int_{N_k}\frac{\pi\ctg\pi z}{z^2}\dz? \) What identity results if we let \(\displaystyle k\to\infty\)? Difficulty: 5. |
Problem 1808
\(\displaystyle \frac1{1^3}-\frac1{3^3}+\frac1{5^3}-\frac1{7^3}+\frac1{9^3}-\frac1{11^3}+-\ldots = ? \) Difficulty: 6. Solution is available for this problem. |
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