The summation challenge evaluates the infinite series $$\sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k}$$, which converges to the value $$\frac{\pi}{4} + \frac{1}{2}\ln(2)$$. The series is broken down into two components: an alternating series of odd-indexed terms and a scaled alternating series of even-indexed terms. The analysis demonstrates the application of series convergence techniques and the use of known mathematical constants.
PREREQUISITESMathematicians, students studying calculus or analysis, and anyone interested in advanced series evaluation techniques.
Opalg said:[sp]$$\begin{aligned}\sum_{k=1}^{\infty} (-1)^{\left\lfloor \frac{k+3}{2} \right\rfloor} \frac{1}{k} &= 1 + \frac12 - \frac13 - \frac14 + \frac15 + \frac16 - \frac17 -\frac18 + \ldots \\ &= \Bigl(1 - \frac13 + \frac15 - \frac17 + \ldots\Bigr) + \frac12\Bigl(1 - \frac12 + \frac13 - \frac14 + \ldots\Bigr) \\ &= \frac\pi4 + \frac12\ln2\end{aligned}$$[/sp]