2 Infinite Series Solved using Complex Analysis

Here we evaluate a couple of trigonometric series $C=\sum_{k=0}^{\infty}\cos(kx)/(2k+1)$ and $S=\sum_{k=0}^{\infty}\sin(kx)/(2k+1)$ using some beautiful complex analysis via Euler's formula and laurent series expansions leading to closed forms that are definitely real numbers WOW do they look complex, if you know what I mean. Techniques covered: Euler's formula, Maclaurin series of arctangent and the identity $\arctan(iz)=i\artanh(z)$ derived from principal logarithms. My complex analysis lectures:    • Complex Analysis Lectures   If you like the videos and would like to support the channel:   / maths505   You can follow me on Instagram for write ups that come in handy for my videos and DM me in case you need math help: https://instagram.com/maths.505?igshi... My LinkedIn:   / kamaal-mirza-86b380252   Advanced MathWear: https://my-store-ef6c0f.creator-sprin...