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A remarkable formula of Ramanujan

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Ramanujan found the following remarkable formula which relates
π
and
e
to the sum of a generalized continued fraction and a power series, but where neither the continued fraction nor the power series separately relate to
π
and
e
.

or

where
an   − 1 = x (n mod 2), bn = n, n   ≥   1
, and
n!!
is the double factorial.

Sqrt(pi*e/2)

For
x = 1
we get

or

where
an   − 1 = 1, bn = n, n   ≥   1
, and
n!!
is the double factorial. Since it is not known whether
π e
is irrational or not, it is thus not known whether
<div class="@radic@" style="display: inline-block; vertical-align: baseline; margin: -1.5ex 0 0 -0.15em; white-space: nowrap;">2  π e / 2</div>
is transcendental or not (although it is obviously irrational).

Decimal expansion of Sqrt(pi*e/2)

The decimal expansion of
2  π e / 2
is
A059444 Decimal expansion of
2  π e / 2
.
{2, 0, 6, 6, 3, 6, 5, 6, 7, 7, 0, 6, 1, 2, 4, 6, 4, 6, 9, 2, 3, 4, 6, 9, 5, 9, 4, 2, 1, 4, 9, 9, 2, 6, 3, 2, 4, 7, 2, 2, 7, 6, 0, 9, 5, 8, 4, 9, 5, 6, 5, 4, 2, 2, 5, 7, 7, 8, 3, 2, 5, 6, 2, 6, 8, 9, 8, ...}

Continued fraction expansion of Sqrt(pi*e/2)

The simple continued fraction expansion of
2  π e / 2
is
A059445 Continued fraction for square root of
2  π e / 2
.
{2, 15, 14, 1, 2, 3, 17, 1, 1, 5, 1, 30, 1, 3, 2, 1, 1, 1, 3, 3, 1, 4, 2, 9, 2, 1, 9, 1, 7, 1, 6, 1, 5, 1, 5, 3, 1, 1, 3, 1, 36, 4, 18, 2, 1, 2, 4, 1, 3, 366, 3, 1, 1, 16, 2, 1, 2, 2, 1, 3, 3, 1, 5, ...}

Continued fraction part

The continued fraction part is given by

where

is the complementary error function (
erfc
)[1] and
erf (z) = 1 −   erfc (z)
is the error function (
erf
  ).[2]

The continued fraction part has decimal expansion

A108088 Decimal expansion of
1  + 1  /  (1  + 2  /  (1  + 3  /  (1  + 4  /  (1  + 5  /  (1 + ) )
.
{6, 5, 5, 6, 7, 9, 5, 4, 2, 4, 1, 8, 7, 9, 8, 4, 7, 1, 5, 4, 3, 8, 7, 1, 2, 3, 0, 7, 3, 0, 8, 1, 1, 2, 8, 3, 3, 9, 9, 2, 8, 2, 3, 3, 2, 8, 7, 0, 4, 6, 2, 0, 2, 8, 0, 5, 3, 6, 8, 6, 1, 5, 8, 7, 3, 4, ...}

Reciprocal of the continued fraction part

The reciprocal of the continued fraction part is given by

with decimal expansion

A111129 Decimal expansion of the continued fraction
1  + 1  /  (1  + 2  /  (1  + 3  /  (1  + 4  /  (1  + 5  /  (1 + ) )
.
{1, 5, 2, 5, 1, 3, 5, 2, 7, 6, 1, 6, 0, 9, 8, 1, 2, 0, 9, 0, 8, 9, 0, 9, 0, 5, 3, 6, 3, 9, 0, 5, 7, 8, 7, 1, 3, 3, 0, 7, 1, 1, 6, 3, 6, 4, 9, 2, 0, 6, 0, 3, 3, 3, 5, 5, 4, 6, 3, 1, 3, 9, 4, 2, 4, 2, ...}

Power series part

where

is the error function (
erf
  ).[2] The power series part has decimal expansion (which is pretty close to
2  2
= 1.414213562373095
) (see A002193)
A060196 Decimal expansion of
1 + 1 / (1 ⋅  3) + 1 / (1 ⋅  3 ⋅  5) + 1 ⧸ (1 ⋅  3 ⋅  5 ⋅  7) + ...
{1, 4, 1, 0, 6, 8, 6, 1, 3, 4, 6, 4, 2, 4, 4, 7, 9, 9, 7, 6, 9, 0, 8, 2, 4, 7, 1, 1, 4, 1, 9, 1, 1, 5, 0, 4, 1, 3, 2, 3, 4, 7, 8, 6, 2, 5, 6, 2, 5, 1, 9, 2, 1, 9, 7, 7, 2, 4, 6, 3, 9, 4, 6, 8, 1, 6, ...}

See also

Notes

  1. Weisstein, Eric W., Erfc, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Erfc.html]
  2. 2.0 2.1 Weisstein, Eric W., Erf, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/Erf.html]