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.

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