Separation principle

File:Eye of Horus square.png

First six summands drawn as portions of a square.

In mathematics, the infinite series 1/2 + 1/4 + 1/8 + 1/16 + · · · is an elementary example of a geometric series that converges absolutely.

Its sum is

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots =\sum _{n=1}^{\mathrm{\infty }}{\left(\frac{1}{2}\right)}^n=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1.$

Direct proof

As with any infinite series, the infinite sum

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots$

is defined to mean the limit of the sum of the first Template:Mvar terms

$s_n=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots +\frac{1}{2^n}$

as Template:Mvar approaches infinity. Multiplying Template:Mvar by 2 reveals a useful relationship:

$2s_n=\frac{2}{2}+\frac{2}{4}+\frac{2}{8}+\frac{2}{16}+\cdots +\frac{2}{2^n}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots +\frac{1}{2^{n-1}}=1+s_n-\frac{1}{2^n}.$

Subtracting Template:Mvar from both sides,

$s_n=1-\frac{1}{2^n}.$

As Template:Mvar approaches infinity, Template:Mvar tends to 1.

History

This series was used as a representation of one of Zeno's paradoxes. The parts of the Eye of Horus were once thought to represent the first six summands of the series.

See also

• 0.999...
• Dotted note

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Template:Mathanalysis-stub