Can you find root of the equation $$f(k)=1+(1-k^2)\ln(1+\frac{1}{k})?$$ I tried to use Matlab command but it does not give me any result. Can you suggest a method to find root of equation.
2026-03-26 09:45:05.1774518305
Root of function
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As soon as you have the variable both inside and outside log/exp functions, it's hopless to think you can get a closed-form solution in terms of elementary functions.
You can usually get by with only a calculator by using iterative methods. Essentially, just try to express one of the occurences of $k$, put in an initial condition, repeat the process a couple of times and you're there. There are convergence conditions to test that I usually ignore and do it by trial and error (if the values explode to nonsense values, I just pick another $k$ and express that one).
What worked:
$$k=\sqrt{1+(\log (1+1/k))^{-1}}$$
Results: 2 → 1.86 → 1.82376 → 1.81313 → ... → 1.80899