Infinite Series

Sequence and Series are fundamentally different from each other.

• Sequence

A sequence is an arrangement of numbers (or objects). For example,

${3, 7,11,15,19, \cdots}$

is a sequence. The first term is ${a_1 = 3}$, the second term is ${a_2 = 7}$.

In general, ${n}$th term in a sequence is denoted by ${a_n}$.

In a strict mathematical sense, a sequence is a function defined from ${\mathbb N}$ to ${\mathbb R}$, ${\forall n \in \mathbb N}$.

If ${\forall n, a_{n+1} \ge a_n}$, it is said to be monotonically increasing. The sequence mentioned above is monotonically increasing, because ${a_{n+1}= a_n + 4, n>1}$ and ${a_1 = 3}$.

If ${\forall n, a_{n+1} \le a_n}$, it is said to be monotonically decreasing. An example is the sequence of reciprocals of natural numbers.

${1, \frac 1 2 , \frac 1 3, \frac 1 4, \cdots}$

Clearly, ${a_{n+1} < a_n}$.

A sequence, whose members have alternate signs, is an alternating sequence. For example,

${a_{n}=n, if \ n \ is \ odd \ and \ }$

${a_n = -n, if \ n \ is \ even}$

• Convergent and Divergent Sequences

Limit of a sequence, ${L}$ is given by ${\lim \limits_{n \to\infty} a_n}$. In other words, as ${n \to \infty}$, the values ${a_n}$ approach ${L}$.

If the limit is finite, the sequence is said to be convergent. The sequence ${a_n = \frac 1 n, n \in \mathbb N}$ is a convergent sequence, because

${\lim \limits_{n \to \infty} \frac 1 n = 0}$

The famous Fibonacci sequence, given by

${a_1 = 1, a_2 = 1 \ and \ a_n = a_{n-1}+a_{n-2}, \forall n \ge 3}$

is a divergent sequence.

• Bounded Sequences

A sequence is bounded, if

I) there exists a number ${l \ in \mathbb R}$, such that ${a_n \le r}$ OR

II) there exists a number ${g \ in \mathbb R}$, such that ${a_n \ge r}$ OR

III) both I and II are satisfied

A convergent sequence is always bounded, but the converse is not always true. e.g. ${1,-1,1,-1,1,-1, \cdots}$ is bounded, but not convergent. Another example is ${a_n = sin (\frac {n \pi}{4})}$. These are oscillating sequences.

• Series

An (infinite) series is the sum of infinite terms in a sequence. Thus, it takes the form

${a_1 + a_2 + a_3 + a_4 + \cdots = \sum \limits_{n=1}^{\infty} a_n}$

If the sum tends to ${\infty}$ or ${- \infty}$, then the series is divergent. We write,

${\sum \limits_{n \to \infty} a_n = \pm \infty}$

In other words, the limit of sum ${S_n = \sum_n a_n}$ as ${n \to \infty}$ becomes ${\pm \infty}$.

If the limit is a finite number, the series is convergent.

• Testing the Convergence of a Series

It may seem counter-intuitive, but the harmonic series is divergent.

${\sum \limits_{n=1}^{\infty} \frac 1 n = 1+ \frac 1 2 + \frac 1 3 + \frac 1 4 + \frac 1 5 + \cdots = \infty}$

The series of reciprocals of prime numbers is also divergent.

${\frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \frac 1 {11} + \cdots = \infty}$

Thus, knowing the behaviour of a series requires a deeper analysis. (Above results can be proved). People have studied a variety of series and have come up with different tests.

A general approach is analyzing the ${n}$th term of series. If it is ${\ge 1}$, then the series is divergent.

The tests use such common facts to test the behaviour of series.

• Cauchy’s Test (${n}$th Root Test)

Let ${a_n}$ denote the ${n}$th term in the series.

I) Applicable only when there are ONLY positive terms in the series

II) If ${\lim \limits_{n \to \infty} a_n < 1}$, the series is convergent.

III) If ${\lim \limits_{n \to \infty} a_n > 1}$, the series is divergent.

IV) If ${\lim \limits_{n \to \infty} a_n = 1}$, the test fails.

• ${p}$-Series Test

The ${p}$-series is a special case of Riemann’s ${\zeta}$ function, given by

${\zeta (s) = \sum \limits_{n=1}^{\infty} \frac {1}{n^s}}$

for values of ${s}$, whose real part is ${> 1}$. It is defined for other complex numbers using analytic continuation and a functional equation. When ${s}$ is real, it becomes the ${p}$-series.

I) The series is convergent if ${p>1}$ and divergent if ${p \le 1}$.

II) When ${p=1}$, we get the harmonic series.

III) When ${n=2}$, we get ${\frac {\pi^2} 6}$. (Can be obtained by Fourier Series as well)

• Behaviour of Geometric Series

Consider a series with first term ${a}$ and common ratio ${r}$. If ${r \ge 1}$, it is divergent, if ${-1 < r < 1}$, it is convergent.

If ${r \le -1}$, the series is oscillatory.

• Comparison Test

Let ${\sum a_n}$ and ${\sum b_n}$ be 2 series of positive terms. Comparing the 2, if

${\lim \limits_{n \to \infty} \frac {a_n}{b_n}}$

is a non-zero finite number, then they both have identical behaviour.

The series ${b_n}$ is known as the auxiliary series.

• D’Alembert’s Test (Ratio Test)

I) Applicable for series of positive terms

II) If ${\lim \limits_{n \to \infty} \frac {a_n}{a_{n+1}} > 1}$, then the series is convergent.

III) If ${\lim \limits_{n \to \infty} \frac {a_n}{a_{n+1}} < 1}$, then the series is divergent.

IV) If the limit is ${1}$, the test is inconclusive.

• Raabe’s Test

I) Generally applied when D’Alembert’s test fails

II) Take the limit ${\lim \limits_{n \to \infty} n \times (\frac {a_n}{a_{n+1}}}$.

III) Conditions are similar to D’Alembert’s test

• Cauchy’s Condensation Test

I) Useful when ${log}$ terms are present

II) Consider a function ${f(n)}$ which continuously decreases as ${n}$ increases. The behavior of ${\sum \limits_{n=1}^{\infty} a^n f(a^n)}$ decides the behaviour of ${\sum \limits_{n=1}^{\infty} f(n)}$.

III) ${a \in \mathbb Z, a > 1}$.

• Auxiliary Series ${\sum \frac {1}{n (ln \ n )^p}}$

I) ${p>1}$, the series is convergent.

II) ${p\le 1}$, the series is divergent.

• Gauss’ Test

I) Let ${\sum \limits_{n=1}^{\infty} a_n}$ be a series, s.t.

${\frac {a_n}{a_{n+1}} = 1 + \frac l n + \frac {b_n}{n^p}, p > 1, b_n \ is \ bounded}$

Then if ${l >1}$, then the sequence converges, else diverges.

• Logarithmic Test

I) If ${\sum \limits_{n=1}^{\infty} n \ ln \ \frac {a_n}{a_{n+1}} > 1}$, the series ${\sum \limits_{n=1}^{\infty} a_n}$ converges. If the limit is ${< 1}$, the series diverges.

II) Applied when D’Alembert’s test fails

• Bertrand’s Test/ de Morgan’s Test

I) Behavior of ${\sum \limits_{n=1}^{\infty} a_n}$ is governed by

${\lim \limits_{n \to \infty} \Big \{ \Big[ n \times \big(\frac {a_n}{a_{n+1}}-1 \big)-1 \Big] \times ln \ n \Big \}}$

II) Limit ${>1}$, ${\sum \limits_{n=1}^{\infty} a_n}$ converges.

III) Limit ${<1}$, ${\sum \limits_{n=1}^{\infty} a_n}$ diverges.

• Alternative Test

Change the function to

${\lim \limits_{n \to \infty} \Big \{ \Big[ \big(n \times \frac {a_n}{a_{n+1}}-1 \big)-1 \Big] \times ln \ n \Big \}}$

• Cauchy’s Integral Test

Let the ${n}$th term be ${a_n = f(n)}$. If ${\int \limits_1^{\infty} f(x)dx}$ is finite, ${\sum \limits_{n=1}^{\infty} a_n}$ converges. If If ${\int \limits_1^{\infty} f(x)dx}$ is infinite, ${\sum \limits_{n=1}^{\infty} a_n}$ diverges.

• Leibniz Test for Alternating Series

The series ${\sum \limits_{n=1}^{\infty} (-1)^{n-1} a_n}$ is convergent if ${|a_{n+1}|< |a_n|}$ and ${\lim \limits_{n \to \infty} a_n = 0}$.

• Absolute and Conditional Convergence

I) When ${\sum \limits_{n=1}^{\infty} |a_n|}$ converges, the series ${\sum \limits_{n=1}^{\infty} a_n}$ is convergent.

II) If the series of absolute values is divergent, the series ${\sum \limits_{n=1}^{\infty} a_n}$ is conditionally convergent. e.g.

${\sum \limits_{n=1}^{\infty} \frac {(-1)^{n+1}}{n}}$

This series converges to ${ln \ 2}$.

*** It has been proven that a conditionally convergent series can be made to converge to any sum, including ${\infty}$ or ${- \infty}$.

• Power Series

Functions can be expanded in a power series. If the series is convergent, the expansion is ${valid}$.

${f(x) = \sum \limits_{n=0}^{\infty} b_n (x-c)^n}$

Note the constant term ${b_0}$. If for ${l_1 < x < l_2}$, the series is convergent, then ${[l_1, l_2]}$ is the range of convergence.