Lecture 03

Jakob Bernoulli &amp; Convergence Series
An infinite series is an ordered sequence of numbers usually given as a formula representing the nth element.

Infinite series were the subject of mathematical study by the 1660s.

Convergence/Divergence
When we take the sum (or product) of a series, the series is found to converge (arrive at a limiting value) or diverge (grow without bounds).

The mathematician Cauchy stressed the importance of considering the convergent or divergent behaviour of these sums and products of series.

Numerical Series
A numerical series in $$\Re$$:

$$\sum_{i=0}^\infty a i = a_0 + a_1 + a_2 + a_3 + \cdots$$

Functional Series
And this is a functional series:

$$\sum_{i=0}^\infty f_i (x) = f_0 (x) + f_1 (x) + f_2 (x) + f_3 (x) + \cdots$$

Power Series
A power series has the form:

$$\sum_{n=0}^\infty a_n x^n$$

Geometrical Reasoning About Infinite Series
Consider the series:

$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = 1$$

In this case, we have the surprising situation that inspection of a geometrical figure leads to a result about infinite series. The following figure, depicting recursive subdivision of the unit square, makes clear why this series converges to 1:



Newton's Binomial Theorem
Isaac Newton arrived at this theorem in the 1660s:

$$1 + a^n = 1 + n_a + \frac{n(n-1)}{2} \cdot a^2 + \frac{n(n-1)(n-2)}{3!}\cdot a^3 + \cdots$$

Let $$n = \frac{1}{2}$$; then this infinite series, the "Taylor polynomial", gives one the square root of an arbitrary number.

TO DO: insert figure (unit-circle-axes)

$$x^2 + y^2 = 1 \rightarrow y = \sqrt{1-x^2}$$

$$A = \pi$$ (unit circle)

$$\frac{\pi}{4} = \int_0^1 \sqrt{1-x^2}\,dx$$

Newton used the Binomial Theorem, discussed previously, to evaluate $$(1-x^2)^\frac{1}{2}$$ to obtain an infinite series and then perform term-by-term integration.

Newton thus obtained an early estimate of the value of &pi; as the area of the unit circle.

Harmonic Series
The harmonic series, so-named because the values used in the denominators are related to the note values commonly found in musical harmonies, looks like:

$$S = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots$$

This infinite series was studied by Jakob Bernoulli, whose approach was to take partial sums of the series in order to see whether it approached some limited value.


 * adding 83 terms yields ≈ 5
 * adding 1200 terms yields ≈ 10
 * adding 25,000,000 terms yields ≈ 20

Does this series converge or diverge?

$$\frac{1}{n} \rightarrow 0$$ as $$n \rightarrow \infty$$, so intuition suggests that the series converges.

However, that is not the case. This series actually diverges, but very slowly&hellip;

A series converges as the nth term approaches 0, but the converse is not true. [explain further]

Leibniz, who was living in Paris and working as an ambassador in the 1670s, explained the divergence using the "triangular numbers" as follows:

TO DO: insert figure (triangular numbers)

$$T = 1 + \frac{1}{3} + \frac{1}{6} + \frac{1}{10} + \cdots$$

Note: $$1 + 2 + \cdots + k = \frac{k(k+1)}{2}$$

$$\frac{1}{2}T = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \cdots$$