We already know that the irrational number $\pi$ can be expressed in this way:
$
\pi =4-\frac{4}{2}+\frac{4}{5}+\cdots +\left( -1 \right) ^{n+1}\frac{4}{2n-1}+\cdots=\sum\limits_{n=1}^\infty\left( -1 \right) ^{n+1}\frac{4}{2n-1}
$
Can all irrational numbers be expressed by infinite number series?
If so, can any transcendental equation have analytic solutions in the form of series?
$$e^{x}+\sin(x)-3=0$$
Best Answer
$e^x$ and $\sin x$ both have very simple expansions into infinite series (with rational coefficients). So a series for a solution of $e^x+\sin x-3=0$ can be obtained by the technique of reversion of series. See, for example, http://mathworld.wolfram.com/SeriesReversion.html or https://en.wikipedia.org/wiki/Lagrange_inversion_theorem