Engineering Mathematics II
Sequences and Series
Module I: Sequences and Series. This chapter develops the fundamental concepts of limits, convergence, monotonicity, infinite series, convergence tests and standard power series. These ideas form the backbone of mathematical analysis and approximation methods.
1. Sequences
A sequence is a function whose domain is the set of natural numbers. It is written as {a_n} or more commonly as a1, a2, a3 and so on.
- Limit of a sequence:A sequence {a_n} has limit L if the terms get arbitrarily close to L as n increases. Symbolically:
lim (n→∞) a_n = L. - Convergent sequence: has a finite limit.
- Divergent sequence: does not approach a fixed number.
- Bounded sequence: if
m ≤ a_n ≤ M. - Monotonic sequences: always increasing or always decreasing. A monotone and bounded sequence is always convergent.
Example
a_n = 1/n is positive, decreasing and bounded below by 0. Hence it converges to 0.
Diagram: sequence-limit.png
2. Infinite Series
A series is the sum of the terms of a sequence.S = a1 + a2 + a3 + .... Convergence depends on the behavior of partial sums.
- Partial sum:
S_n = a1 + a2 + ... + a_n. - The series converges if S_n approaches a finite limit.
- Necessary condition:
lim (n→∞) a_n = 0.
Standard Convergence Tests
- Comparison Test: compare with known convergent or divergent series.
- Ratio Test:
L = lim |a_(n+1)/a_n|. L less than 1 implies convergence. - Root Test:
L = lim (|a_n|)^(1/n). - Alternating Series Test: decreasing terms with limit 0 converge.
Diagram: series-tests.png
3. Power Series
A power series is of the formΣ a_n (x - c)^n.
- Converges within radius R around x = c.
- Diverges outside R.
- Convergence at boundary points tested separately.
Standard Series Expansions
e^x = 1 + x + x^2/2! + ...sin x = x - x^3/6 + ...cos x = 1 - x^2/2! + ...ln(1+x) = x - x^2/2 + ...1/(1-x) = 1 + x + x^2 + ...for |x| less than 1
Diagram: taylor-series.png