I plugged in the following series into a calculator: $$\sum_{n=1}^\infty \ln(1+\frac{1}{n^2})$$ and got a result of approximately $1.29686$. That's nice and all, but I want to know: is this result irrational or perhaps even transcendental? I know that the natural log of any rational number is transcendental, therefore every single term in the series is transcendental, but adding together transcendental numbers infinitely may yield rational sums, e.g. $$\sum_{n=0}^\infty \frac{(-1)^n}{\pi(2n+1)} = 1/4$$ So how can I prove that my result, ~$1.29686$, is transcendental?
2026-04-02 16:00:14.1775145614
How to tell if an infinite series sum will be rational or irrational?
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