The General Binomial Theorem: Part 2

In the previous post we established the general binomial theorem using Taylor's theorem which uses derivatives in a crucial manner. In this post we present another approach to the general binomial theorem by studying more about the properties of the binomial series itself. Needless to say, this approach requires some basic understanding about infinite series and we will assume that the reader is familiar with ideas of convergence/divergence of an infinite series and some of the tests for convergence of a series.

The General Binomial Theorem: Part 1



One of most basic algebraic formulas which a student encounters in high school curriculum is the following $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$ and its variant for $(a - b)^{2}$. And after many exercises and problems later one encounters another formula of similar nature namely $$(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$$ and one wonders if there are similar formulas for higher powers of $(a + b)$.

Theories of Circular Functions: Part 3

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Continuing our journey from last two posts we present some more approaches to the development of the theory of circular functions. One approach is based on the use of infinite series and requires basic knowledge of theory of infinite series. This approach is particularly well suited for treating circular functions as functions of a complex variable, but we will limit ourselves to the case of real variables only.

Theories of Circular Functions: Part 2

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In the last post we covered the traditional approach towards the theory of circular functions which is based on geometric notions related to a circle. In my opinion this approach is the easiest to understand and therefore commonly described in almost any trigonometry textbook (but without the theoretical justification of length (and area) of arcs (and sectors). However it is interesting to also have an approach which is independent of any geometrical notions. In this post we will introduce the circular functions as inverses to certain integrals.

Theories of Circular Functions: Part 1

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While answering certain questions on MSE in last few weeks it occurred to me that ample confusion is prevalent among students (and instructors alike) regarding a theoretically sound development of circular (or trigonometric) functions. In the past I had hinted at two usual approaches to trigonometry, but I guess that was not enough and hence I am writing this series on the development of circular functions (like I did for the exponential and logarithmic functions earlier).