$\tan(x) = x$. Find the values of $x$

sequences-and-series trigonometry

There is no closed form for the solutions of $ \tan(x) = x $, but let me state a few interesting facts. Let $ (\lambda_{n})_{n \in \mathbb{N}} $ be the sequence that lists the positive solutions of $ \tan(x) = x $ in increasing order. Then

  • $ \displaystyle \sum_{n=1}^{\infty} \frac{1}{\lambda_{n}} = \infty $.

  • $ \displaystyle \sum_{n=1}^{\infty} \frac{1}{\lambda_{n}^{2}} = \frac{1}{10} $.


I gave a talk on this equation in April 2006 and (a .pdf version of) the LaTeX slides I used may be of interest to you.

See also the math StackExchange question Derivation of asymptotic solution of $\tan (x)=x$. Finally, see the posts in this January 2006 sci.math thread: Regarding tan(x) = x.

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