I’ve previously discussed various aspects of the “trigonometric harmonic series” $\sum_{n=1}^\infty\cos n/n$, and in particular showed that the series is conditionally convergent. However, we haven’t found the actual value it converges to; our argument only shows that the value must be smaller than about $2.54$ in absolute value. In this post, I’ll give a closed-form expression for the exact value that the series converges to.

# The value of the trigonometric harmonic series

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