Abstract
In this work, we give two new Taylor expansions of arctan(x + ω), where ω represents a finite increment of x. We discover several remarkable infinite series from these expansions by special substitutions. Some of these infinite series give BBP-type formulae.
Highlights
In 1673, Leibniz obtained an elegant infinite series of π/4 that bears his name [1], since this field of mathematic began to attract widespread attentions among mathematicians
We find several new infinite series from these expansions, and their numerical results are BBP-type formulas
We propose two new Taylor expansions of arctan(x + ω)
Summary
In 1673, Leibniz obtained an elegant infinite series of π/4 that bears his name [1], since this field of mathematic began to attract widespread attentions among mathematicians. The numerical method is not priori because it can provide only the formula but not the origin of the formula For this reason, when a BBP-type formula is discovered via an algorithm, we still need to provide a mathematical proof to make the result rigorous. The second method to find a BBP-type formula is based on the theory of calculus and infinite series. We find several new infinite series from these expansions, and their numerical results are BBP-type formulas. Let y = arctanx, if x changes by a finite increment ω, according to the well-known Taylor’s expansion, we can represent the value of y at x + ω as arctan(x This series, since the expression has no certain law, can provide nothing valuable to us. Suppose ω is an arbitrary finite increment of the variable x, arctan(x n=1
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