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

Entry 1.

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.

Entry 2.

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.
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