Discovering a Family of Machin-Type Formulas through Finite Series

Entry 1.

For each positive integer n,
.
 

Formula (I) demonstrates that for each value of n, a Machin-type formula is obtained. As n increases, an infinite number of formulas with this structure emerge, forming a family of Machin-type formulas, as shown 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.

 

Entry 2.

Another beautiful identity is used to generate many Machin-type formulas. This 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 for n = 0, 1, 2, 3, ..., infinity. 

Collection of Machin-type formulas 

References

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