Mathematics · Ch 5 — Binomial Theorem, Sequences and Series

Introduction

What Binomial Theorem Is For

Binomial theorem facilitates the algebraic expansion of $(a+b)^n$ for a positive integer exponent $n$ . It is used across every branch of mathematics and in the other sciences too — from finding the coefficient of $x^{20}$ in $(2x-7)^{23}$ instantly (rather than multiplying $(2x-7)$ by itself 23 times), to working out the maturity amount on a sum deposited at compound interest for several years, to projecting a country's population a few years ahead from its present size and growth rate, to computing probabilities in a finite sample space where every outcome is either a success or a failure.

The coefficients that appear in the expansion of $(a+b)^n$ (for $n\in\mathbb N$ ) are called binomial coefficients.

Note

A little history. The Greek mathematician Euclid recorded the special case of the binomial theorem for exponent $2$ . The case for exponent $3$ was known in India by the 6th century. In 1544, the German mathematician Michael Stifel introduced the term binomial coefficient and expressed $(1+x)^n$ in terms of $(1+x)^{n-1}$ .

Johann Carl Friedrich Gauss (1777–1855), the "Prince of Mathematics," contributed to number theory, physics and astronomy — number theory was his own favourite field, which he called the "Queen of Mathematics." He is remembered here for the schoolboy anecdote of instantly summing $1+2+\cdots+100=5050$ when his teacher set the class this task to keep them occupied; nobody is certain which method the young Gauss actually used.

Sequences and Series: A Second Thread

Sequences and series problems have occupied mathematicians for over a thousand years. One famous legend concerns the invention of chess: the number of grains owed for each successive square of the board follows $1, 2, 4, 8, \ldots$ — doubling every square, so that the 64th square alone demands an almost unimaginable number of grains. Arithmetic and geometric progressions like this show up in many real-life situations.

Roughly speaking, a sequence is an arrangement of objects in a definite order, and a series is the sum of the terms of a sequence of numbers. The idea of an infinite series lets us compute quantities such as $\sin\left(\tfrac{9\pi}{44}\right)$ , $\log 43$ , or $e^{20}$ to any desired level of accuracy — sequences and series are foundational to differential equations and analysis as well.

This chapter has two connected halves: binomial theorem and its applications, and sequences and series — arithmetic, geometric, arithmetico-geometric, harmonic, and their infinite counterparts, culminating in the binomial series for a rational exponent, the exponential series, and the logarithmic series.

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