(April 09, 2009)

Some finite series help to find a family of Machin-type formula.

For each positive integer n,

.

(I) illustrates that for each value of n, it gives a Machin-like formula. Therefore, many Machin-like formulas of this similar structure type are generated as n tend to infinity. This establishes a *family of Machin-like formula that* is depicted in the following example.

* *

Example.

1. For n = 0, (I) gives

.

2. For n = 1, (I) gives

.

3. For n = 2, (I) gives

.

4. For n = 3, (I) gives

.

And so on.

Another beautiful identity is used to generate many Machin-type formulas of a class of identities involving a well known formula that Hwang Chien-Lih (1997) used to compute the digits of Pi [1], namely

For each positive integer n,

.

Many Machin-like formulas of (II) are generated as n = 0, 1, 2, 3, ..., infinity.

Collection of Machin-Like formulas

**References**

[1] Chien-Lih Hwang, More Machin-Type Identities, The Mathematical Gazette, 1997, p. 120-121.

.

Tue N. Vu, Machin-Type Formula, 04/09/2009, from Series Math Study Resource.

Hyperlink: http://seriesmathstudy.com/sms/machintypetv.

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