

A196504


Decimal expansion of the least x > 0 satisfying x + tan(x) = 0.


11



2, 0, 2, 8, 7, 5, 7, 8, 3, 8, 1, 1, 0, 4, 3, 4, 2, 2, 3, 5, 7, 6, 9, 7, 1, 1, 2, 4, 7, 3, 4, 7, 1, 4, 3, 7, 6, 1, 0, 8, 3, 8, 0, 0, 2, 8, 7, 5, 9, 3, 9, 4, 0, 8, 8, 8, 1, 7, 1, 6, 6, 0, 7, 4, 4, 4, 9, 8, 6, 6, 5, 0, 3, 1, 0, 4, 2, 7, 6, 2, 3, 4, 5, 9, 2, 2, 7, 9, 5, 1, 5, 0, 4, 2, 5, 6, 3, 0, 6, 3, 9
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OFFSET

1,1


COMMENTS

Let L be the least x > 0 satisfying x + tan(x) = 0.
Then L is also the least x > 0 satisfying x = (sin(x))(sqrt(1+x^2)).
Consequently, for 0 < x < L, for all p > 0, 1/sqrt(1+x^2)  1/x^p < sin(x) < 1/sqrt(1+x^2) for 0 < x < L.
See A196500A196503 and A196505 for related constants and inequalities.
The number L also occurs in connection with Du Bois Reymond's constants; see the Finch reference.
For x = L the area of right triangle with vertices (0,0), (x,0) and (x,sin(x)), i.e., the one inscribed into the halfwave curve, is maximal.  Roman Witula, Feb 05 2015


REFERENCES

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, page 239.


LINKS

Table of n, a(n) for n=1..101.


EXAMPLE

L = 2.02875783811043422357697112473471437610838002...
1/L = 0.4929124517549075741877801898222329769156970132...


MATHEMATICA

Plot[{Sin[x], x/Sqrt[1 + x^2]}, {x, 0, 9}]
t = x /.FindRoot[Sin[x] == x/Sqrt[1 + x^2], {x, .10, 3}, WorkingPrecision > 100]
RealDigits[t] (* A196504 *)
1/t
RealDigits[1/t] (* A196505 *)


CROSSREFS

Cf. A196505, A196500, A196502.
Sequence in context: A021497 A201735 A029593 * A182550 A004514 A278748
Adjacent sequences: A196501 A196502 A196503 * A196505 A196506 A196507


KEYWORD

nonn,cons


AUTHOR

Clark Kimberling, Oct 03 2011


STATUS

approved



