General Inverse Tangent Series of Unknown Names

(Posted 08/02/2008)

The following general finite series formulas of unknown names have been established in connecting to the inverse tangent.

· For any positive integer,

,

whereand. If the value of y is outside of the range, addin the left side of (I), where m is integer.

Rewrite (I) in the expanded form

.

· Using the complex property tanh(ix) = tan(x) and tan(ix) = itanh(x) to the formula (I), gives

, where,and.

Rewrite (II) in the expanded form

.

· Similarly, using tan(ix) = itanh(x) and substitute it into (II), gives

,

where,and x > y.

Rewrite (III) in the expanded form

.

· Using only tanh(ix) = itan(x) and substitute it into (I), gives

,

where, x > y,,, and i is integer.

Rewrite (IV) in the expanded form

.

· The formulas (I), (II), (III), and (IV) can be used to derive innumerable series of variety forms. For instance, we obtain four Machin-like formulas when considering n = 1.

· If we substitute y = pi/2 in (V), we obtain

Or

Rewrite (VI) in terms of tanh x/2 by using the identity.

.

· By setting x = tanh x/2 in (VII), we obtain the Machin-like formula

.

Similarly, we get another formula for x < 0,

.

(Posted August 2, 2008)

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