Sequence and Series are fundamentally different from each other.
A sequence is an arrangement of numbers (or objects). For example,
is a sequence. The first term is , the second term is .
In general, th term in a sequence is denoted by .
In a strict mathematical sense, a sequence is a function defined from to , .
If , it is said to be monotonically increasing. The sequence mentioned above is monotonically increasing, because and .
If , it is said to be monotonically decreasing. An example is the sequence of reciprocals of natural numbers.
A sequence, whose members have alternate signs, is an alternating sequence. For example,
Convergent and Divergent Sequences
Limit of a sequence, is given by . In other words, as , the values approach .
If the limit is finite, the sequence is said to be convergent. The sequence is a convergent sequence, because
The famous Fibonacci sequence, given by
is a divergent sequence.
A sequence is bounded, if
I) there exists a number , such that OR
II) there exists a number , such that OR
III) both I and II are satisfied
A convergent sequence is always bounded, but the converse is not always true. e.g. is bounded, but not convergent. Another example is . These are oscillating sequences.
An (infinite) series is the sum of infinite terms in a sequence. Thus, it takes the form
If the sum tends to or , then the series is divergent. We write,
In other words, the limit of sum as becomes .
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.
The series of reciprocals of prime numbers is also divergent.
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 th term of series. If it is , then the series is divergent.
The tests use such common facts to test the behaviour of series.
Cauchy’s Test (th Root Test)
Let denote the th term in the series.
I) Applicable only when there are ONLY positive terms in the series
II) If , the series is convergent.
III) If , the series is divergent.
IV) If , the test fails.
The -series is a special case of Riemann’s function, given by
for values of , whose real part is . It is defined for other complex numbers using analytic continuation and a functional equation. When is real, it becomes the -series.
I) The series is convergent if and divergent if .
II) When , we get the harmonic series.
III) When , we get . (Can be obtained by Fourier Series as well)
Behaviour of Geometric Series
Consider a series with first term and common ratio . If , it is divergent, if , it is convergent.
If , the series is oscillatory.
Let and be 2 series of positive terms. Comparing the 2, if
is a non-zero finite number, then they both have identical behaviour.
The series is known as the auxiliary series.
D’Alembert’s Test (Ratio Test)
I) Applicable for series of positive terms
II) If , then the series is convergent.
III) If , then the series is divergent.
IV) If the limit is , the test is inconclusive.
I) Generally applied when D’Alembert’s test fails
II) Take the limit .
III) Conditions are similar to D’Alembert’s test
Cauchy’s Condensation Test
I) Useful when terms are present
II) Consider a function which continuously decreases as increases. The behavior of decides the behaviour of .
I) , the series is convergent.
II) , the series is divergent.
I) Let be a series, s.t.
Then if , then the sequence converges, else diverges.
I) If , the series converges. If the limit is , the series diverges.
II) Applied when D’Alembert’s test fails
Bertrand’s Test/ de Morgan’s Test
I) Behavior of is governed by
II) Limit , converges.
III) Limit , diverges.
Change the function to
Cauchy’s Integral Test
Let the th term be . If is finite, converges. If If is infinite, diverges.
Leibniz Test for Alternating Series
The series is convergent if and .
Absolute and Conditional Convergence
I) When converges, the series is convergent.
II) If the series of absolute values is divergent, the series is conditionally convergent. e.g.
This series converges to .
*** It has been proven that a conditionally convergent series can be made to converge to any sum, including or .
Functions can be expanded in a power series. If the series is convergent, the expansion is .
Note the constant term . If for , the series is convergent, then is the range of convergence.