Analysis

Analysis is a wider term then calculus but is sometimes taken to be equivalent. It encompasses calculus, infinite series, measure theory (developed by Henri Lebesgue in 1901), a part of set theory, etc. In the past, it was about the infinite and the infinitesimal, now it's given a clear definition in terms of limits and continuity.

Infinite series is the sum of an infinite sequence of numbers. It can be convergent, divergent or conditionally convergent. Convergent series tend to a finite number, while divergent ones tend to infinity or oscillates between different numbers. An example of the latter is the series S = 1 - 1 + 1 - 1 + 1 - 1 + ... By arranging in two different methods we get S = 0 and 1 respectively: S = (1 - 1) + (1 - 1) + (1 - 1) + ... = 0 and S = 1 + (- 1 + 1) + (- 1 + 1) + ... = 1. A conditionally convergent series is one in which the convergence is obtained by arranging the terms in a specific manner, arranged in some other manner renders the series infinite.

Some convergent series:

S₁ = 1 + ¹/₂ + ¹/₄ + ¹/₈ + ... + (¹/₂)^n-1 + ...

S₂ = 1 - ¹/₂ + ¹/₄ - ¹/₈ + ... + (-¹/₂)^n-1 + ...

S₁ is a geometric series with first term a = 1 and common ratio r = ¹/₂. There is a formula in geometric series analysis which says that S = ^a/_(1-r), hence S₁ = 2. Using the same formula, this time with r = -¹/₂, we have S₂ = ²/₃.

The above series can all be re-written as a sequence containing the sum to nth term. E.g. S₁ can be re-written as the sequence 1, 3/2, 7/4, 15/8, ..., (2ⁿ-1)/2^n-1, ... As n tends to infinity, the sequence tends to 2. Consider the following sequences:

S₃ = ¹/₁ ,³/₂ ,⁷/₅ ,¹⁷/₁₂ ,...,^a/_b ,^(a+2b)/_(a+b) , ...

S₄ = ⁴/₁ ,⁸/₃ ,³²/₉ ,¹²⁸/₄₅ ,... where the nth term is obtained by multiplying the (n-1)th term by n/(n+1) if n is even, and by (n+1)/n if n is odd.

S₅ = 1¹,(³/₂)² ,(⁴/₃)³ ,(⁵/₄)⁴ ,...,[^(n-1)/_(n)]ⁿ , ...

S₃, S₄ and S₅ tends to √2, π and e respectively.

An example of a conditionally convergent series:

S₆ = 1 - ¹/₂ + ¹/₃ - ¹/₄ + ...

Adding the terms from left to right gives ln 2, but re-arranging as (1 - ¹/₂) - ¹/₄ + (¹/₃ - ¹/₆) -¹/₈ + ... = ¹/₂ - ¹/₄ + ¹/₆ - ¹/₈ = ¹/₂ (1 - ¹/₂ + ¹/₃ - ¹/₄ + ...) = ¹/₂ ln 2.

The harmonic series is the sum of the reciprocal of all the natural numbers. It's a special case of the Riemann zeta function ζ(s), where s = 1.

It's the most marginally divergent series in the Riemann zeta function, i.e. for s > 1, ζ(s) is convergent, but for s ≤ 1, ζ(s) is divergent. In fact, let ζ_n(s) denote the sum to n terms, then ζ_n(1) ~ ln n. It is known that ln x < x^ε however small ε is. The harmonic series is encountered many times in Mathematics. E.g. what is the maximum overhang one can achieve with a standard deck of 52 cards? The answer is ¹/₂ (1 + ¹/₂ +¹/₃ + ... + ¹/₅₁) = ¹/₂ ζ₅₁(1). To prove the divergence of the series, observe that ¹/₃ + ¹/₄ > ¹/₂, ¹/₅ + ¹/₆ + ¹/₇ + ¹/₈ > ¹/₂ and so on, then adding up the series is like adding up infinitely many terms each greater than ¹/₂, hence the sum is infinite.

Another series that diverges is the sum of the reciprocals of all primes. It diverges much slower than the harmonic series.

How about the series ζ(2)? The solution is π²/6. This is due to Euler who gave us a method that works for all even n. E.g. ζ(4) = π⁴/90, ζ(6) = π⁶/945, ..., ζ(26) = 1315862π²⁶/11094481976030578125. No one knows if there's a closed form solution for odd powers though.

Euler re-wrote the zeta function in the following form:

By observing equations (1) and (3), we have ∑ n^-s = ∏ (1-p^-s)^-1. This is the Euler Product Formula. He obtained it by the following steps:

Looking at (1), it is clear that the series diverges for all s ≤ 1. However, via some domain stretching manipulation, we can allow the zeta function to have values for all s except s = 1. First we consider the range s between 0 and 1. Let η(s) be define below.

η(s) is convergent. With this we can re-write (1) as ζ(s) = (1-2^s-1)^-1η(s). And since η(s) is defined for s > 0, so is ζ(s). Next consider s ≤ 0. Euler proposed the following formula in 1749. Riemann proved it in the 1859.

Since ζ(s) is defined for s > 1, ζ(s) is defined for s < 0. (10) is zero only when sin[π(1-s)/2] is zero, i.e. when 1 - s = -2, -4, -6, ... These are the trivial zeros of the zeta function.

The M�bius function μ(n) where n is a natural number is defined as below:

1. μ(1) = 1

2. μ(n) = 0 if n has a square factor

3. μ(n) = -1 if n is a prime or the product of odd number of distinct primes

4. μ(n) = 1 if n is the product of even number of distinct primes

With this function, the reciprocal of ζ(s) can be written as 1 / ζ(s) = ∑ [μ(n) / n^s] as follows

There is a use for this. What is the probability that a natural number N chosen at random is not divisible by any nth power? Answer: 1 / ζ(n).

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