Series

If we add the terms of an infinite sequanece {a_n}, we get an expression of the form

a₁ + a₂ + ... + a_n

which is called an infinite series (or just series), and is denoted by the formal expression

\sum_{n = 0}^{\infty} a_{n} 3.1.1

We consider the partial sums

s₀ = a₀
s₀ = a₀ + a₁
s₀ = a₀ + a₁ + a₂
s_n = a₀ + a₁ + ... + a_n

and, in general,

s_{n} = a_{1} + \dots + a_{n} = \sum_{i = 0}^{n} a_{i}

These partial sums form a new sequence {s_n}. If lim_{n ⟶ ∞} s_n = s exists (as a finite number), then we call it the sum of the infinite series Σ a_n.

Definition 3.1.1 With respect to series 3.1.1 considering the limit lim_{n ⟶ ∞} s_n, we can have the following situations:

If the limit exists and is finite, we say that the series converges; The value s of the limit is called sum of the series, and we write:
$\sum_{n = 0}^{\infty} a_{n} = lim_{n \to \infty} \sum_{i = 0}^{n} a_{i} = lim_{n \to \infty} s_{n} = s$
If the limit is ±∞ we say the series diverges;
If the limit does not exist, we say the series is indeterminate. □

The basic necessary and sufficient condition for convergence, due to Cauchy, is a simple reformulation in terms of series of Cauchy's criterion' for convergence of a sequence, which we introduced for sequences. We simply have to refer to the sequence of partial sums {s_n} of an infinite series Σ a_n, giving the following criterion for series.

Theorem 3.1.2 (Cauchy’s convergence criterion for series). Let Σ a_n be an infinite series with a sequence of partial sums {s_n}. The series converges if and only if its sequence {s_n} is a Cauchy sequence.

Proof. Suppose the Cauchy criterion holds. Then for every ε > 0, there is an integer N ∈ ℕ such that

n > m \geq N \Rightarrow | s_{n} - s_{m} | = | \sum_{k = 0}^{n} a_{k} - \sum_{k = 0}^{m} a_{k} | = | a_{m + 1} + \dots + a_{n} | < ε

Thus the sequence {s_n} is Cauchy in the sense discussed for sequences. We conclude that the sequence {s_n} converges; by definition, therefore, the series converges.
Conversely, if the series converges then, by definition 3.1.1, the sequence {s_n} of partial sums converges. In particular, the sequence {s_n} must be Cauchy. Thus, for any ε > 0, there is a number N > 0 such that

n ≥ m < N ⟶ |s_n − s_m| < ε

This just says that

| \sum_{k = m + 1}^{n} a_{k} | < ε 3.1.2

and this last inequality is the Cauchy criterion for series. □

From the theorem a simple necessary condition for convergence follows. This condition does not guarantee the convergence of the series, but if not satisfied it excludes it.

Corollary 3.1.3. (The zero test) If the series Σ a_n is convergent, then lim_{n ⟶ ∞} a_n = 0

Proof. Since we are assuming that the series converges, it must satisfy Cauchy criterion. Let ε > 0. Then there is an integer N ≥ 1 such that if n ≥ m > N, then 3.1.2 holds; If we take n = m and m > N, 3.1.2 becomes

|a_m| < ε

which proves the claim. □

Cauchy criterion can be states succinctly as: an infinite series converges if and only if it is Cauchy sequence.

Example 3.1.4. The harmonic series is defined by

\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \dots

Using Cauchy's criterion one deduces that the harnomic series diverges. Indeed, the sequence {s_n} if its partial sums verifies

S_{2 n} - S_{n} = \frac{1}{n + 1} + \frac{1}{n + 2} + \dots + \frac{1}{2 n} > \frac{n}{2 n} = \frac{1}{2}

thus the Cauchy criterion fails for any ε < 1/2; the harmonic series diverges. ■

The divergence of the harmonic series was first established by the Italian mathematician Mengoli in 1650.

Example 3.1.3 (Mengoli's series)

\sum_{n = 1}^{\infty} \frac{1}{n (n + 1)}

Observe that 1/n(n +1) = 1/n − 1/(n+1), we are able to give a simpole expression of the sequnce s_n

s_{n} = \sum_{k = 1}^{n} \frac{1}{k (k + 1)} = (1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + \dots + (\frac{1}{n} - \frac{1}{n + 1}) = 1 - \frac{1}{n + 1}

By noting that equal terms of opposite sign can be coupled and vanish. Thus s_n ⟶ 1, hence the series converges and has sum equal to 1. ■

Example 3.1.2. (Geometric progression and series). A sequence is called a geometric progression if each term in the sequence (after the first) is a constant multiple of the previous term.

a, aq, aq₂, aq³, ... , aqⁿ

where a is said first term, and q said common ratio

Thus, if the terms are {a_i} for i = 0, 1, 2, 3..., then a_i+1 = qa_i for some constant, . For example

Progression	Common ratio
1,3,9,27,81, ...	3
16/27, −8/9, 4/3, −2, ...	−3/2
x, x², x³, x⁴ ...	x

The nth term of a geometric sequence whose first term is a and common ratio is a non-zero number q is

a_n = aqⁿ⁻¹

A geometric series is any series that can be put into the form

\sum_{n = 0}^{\infty} a q^{n} = a + a q + a q^{2} + \dots + a q^{n} + \dots

with q ∈ ℝ. The partial sums of a geometric series with q = 1, then s_n = a + a + ... + a = a(n + 1); as n ⟶ ∞ the series diverges in this case.

If instead q ≠ 1, we have

s_n = a + aq + aq² + ... + aqⁿ + ...

and

qs_n = qa + aq² + aq³ + ... + aqⁿ⁺¹ + ...

Subtracting these equations, we get

s_n − qs_n = a − aqⁿ⁺¹

Thus

s_{n} = \sum_{k = 0}^{n} a q^{k} = a + a q + a q^{2} + \dots + a q^{n} = a \frac{1 - q^{n + 1}}{1 - q}

see the exercise on mathematical induction for a rigorous proof of the previous relation.

Considering the limit for n ⟶ ∞ of s_n, we have

\sum_{n = 0}^{\infty} a q^{n} = lim_{n \to \infty} s_{n} = {\begin{cases} \frac{a}{1 - q} if | q | < 1 \\ + \infty if q \geq 1 \\ it doesn't exist if q \leq - 1 \end{cases}

Definition 3.1.3. A decimal expansion of a real number a > 0 is a series of the form

where a₀ is called the integral part of a, and r = 0.a₁a₂a₃... is called its fractional part. The integers 0 ≤ a_k ≤ 9 are called the decimal digits of a. □

Thus we have a geometric series with a = 1 and q = 1/10 that we know to converge. Therefore, each real number a > 0 has a well defined decimal expansion of the form 3.1.3. Clearly, the same argument applies if a < 0.

As it sufficies to work with the fractional part of a > 0, we can assume that a ∈ [0, 1). We say that a = 0.a₁a₂a₃... is a finite decimal (or a terminating decimal) if, for some n, a_k = 0, for k > n ≥ 1; and we say that a is an infinite deciaml with repeating decimals if, except for finitely many initial digits, a block of digits keepts repeating infinitely many times. That is,

In this case, we write a = 0.a₁a₂ ⋅⋅⋅ b₁b₂b_m, and call it a periodic decimal. A terminating decimal can be viewed trivially as an infinite (non-terminating) decimal by attaching an infinite string of 0's at the tail.

The convergence of a goemtric series can be used to express the fact that the periodic decimal 0.9̄ and the number 1 are taken to be the same.

The decimal expansion 0.5 and 0.49̄ represents the same rational number 1/2. Also

Example 3.1.3 Write the number 5.312 = 5.3121212 ... as a ratio of integers.

After the first term we have a geometric series with a = 12/10³ and q = 1/10². Therefore using the formula for the sum of a convergent geometric series