Catalan's Constant - Quickly Converging Series

Quickly Converging Series

The following two formulas involve quickly converging series, and are thus appropriate for numerical computation:


\begin{align}
G & =
3 \sum_{n=0}^\infty \frac{1}{2^{4n}}
\left(
-\frac{1}{2(8n+2)^2}
+\frac{1}{2^2(8n+3)^2}
-\frac{1}{2^3(8n+5)^2}
+\frac{1}{2^3(8n+6)^2}
-\frac{1}{2^4(8n+7)^2}
+\frac{1}{2(8n+1)^2}
\right) \\
& {}\quad -2 \sum_{n=0}^\infty \frac{1}{2^{12n}}
\left(
\frac{1}{2^4(8n+2)^2}
+\frac{1}{2^6(8n+3)^2}
-\frac{1}{2^9(8n+5)^2}
-\frac{1}{2^{10} (8n+6)^2}
-\frac{1}{2^{12} (8n+7)^2}
+\frac{1}{2^3(8n+1)^2}
\right)
\end{align}

and

The theoretical foundations for such series is given by Broadhurst (the first formula) and Ramanujan (the second formula). The algorithms for fast evaluation of the Catalan constant is constructed by E. Karatsuba.

Read more about this topic:  Catalan's Constant

Famous quotes containing the words quickly, converging and/or series:

    How quickly nature falls into revolt
    When gold becomes her object!
    William Shakespeare (1564–1616)

    It might become a wheel spoked red and white
    In alternate stripes converging at a point
    Of flame on the line, with a second wheel below,
    Just rising, accompanying, arranged to cross,
    Through weltering illuminations, humps
    Of billows, downward, toward the drift-fire shore.
    Wallace Stevens (1879–1955)

    I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short series of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.
    Ralph Waldo Emerson (1803–1882)