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Et harum scientiarum porta et clavis est Mathematica

"Mathematics is the door and key to the sciences" - Roger Bacon (~1214-1292)

$$ \sum_{n=1}^\infty\frac{1}{n^z} = \prod_{p \text{ prime}} \frac{1}{1-p^{-z}} $$


4h
comment FindRoot in range
If for some reasons you want to work with FindRoot (e.g. because it is much faster for a large class of transcendental equations) you may exploit the technique using RootIntervals as in this answer First positive root and instead of Sin[x] use the first terms of its Taylor series e.g. N@First@RootIntervals[Series[Sin[x], {x, 0, 15}] // Normal]. This information is all you may need to work out the solution.
4h
comment Finding the parameter that makes real part of an eigenvalue of a matrix 0
@obareey It seems that quite similar issue I discussed here (see also the links I gave there). Nonetheless such problems cannot be justified thoroughly as far as we don't access to the source code. I'll look on this monday if I haven't overlooked anything important (and possibly edit my answer), however it is obvoius that these eigenvalues form the complete set of them since the characteristic equation is of the third order and we can easily verify the solutions.
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revised Solving for a charge distribution from a dipole moment
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answered Finding the parameter that makes real part of an eigenvalue of a matrix 0
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awarded  Nice Answer
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reviewed Looks Good Finding all instances of length $N$ common subwords in two strings
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comment How can I calculate the limit without using the L'Hopital's rule
@MichaelE2 I like it i.e. geometric reasoning.
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reviewed Close How to get integer/rational and real eigenvectors to be the same?