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Suppose the sum \[\sum_{n=1}^\infty \left[ H_n - \gamma - \ln n - \dfrac{\zeta(2n)}{2n} \right]\] can be expressed in the form ...

Find the sum of all solutions to the equation

\[ \large (x^2+5x+5)^{x^2-10x+21}=1 .\]

\[ \frac{ 1 } { 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} +\frac{ 1} {3 \cdot 4\cdot 5\cdot 6\cdot 7} + \frac{ 1}{ 5 \cdot 6 \cdot 7 \cdot 8 \cdot9 } + \cdots = \, ? \]

\[ \large \lim_{n \to\infty} \sum_{m=1}^{n} \dfrac{1}{m+n} =\, ? \]

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