Chapter 02 - Sequence and Series

Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing

Bernoulli’s Inequality

Bounded Sequence

Convergence of the Sequence

Theorem: A convergent sequence of real number has one and only one limit (i.e. limit of the sequence is unique.)

Cauchy Sequence

Theorem: A Cauchy sequence of real numbers is bounded.

Divergent Sequence

Theorem: If $s_n<u_n<t_n$ for all $n\ge n_0$ and if both the $\{s_n\}$ and $\{t_n\}$ converge to same limits as s, then the sequence $\{u_n\}$ also converges to s.

Theorem: If the sequence $\{s_n\}$ converges to s then $\exists$ a positive integer such that $\left| {\,{s_n}}\right|>\frac{1}{2}s$.

Theorem: Let a and b be fixed real numbers if $\{s_n\}$ and $\{t_n\}$ converge to s and t respectively. Then (i) $\left\{a{s_n}+b{t_n}\right\}$ converges to $as+bt$. (ii) $\left\{{s_n}{t_n}\right\}$ converges to st. (iii) $\left\{\frac{{{s_n}}}{{{t_n}}} \right\}$ converges to $\frac{s}{t}$ provided ${t_n}\ne 0\,\,\,\,\forall \,\,\,n$ and $t\ne 0$.

Theorem: For each irrational number x, there exists a sequence $\left\{{r_n}\right\}$ of distinct rational numbers such that $\lim_{n\to \infty}{r_n}=x$.

Let a sequence $\{s_n\}$ be a bounded sequence. (i) If $\{s_n\}$ is monotonically increasing then it converges to its supremum. (ii) If $\{s_n\}$ is monotonically decreasing then it converges to its infimum.

Recurrence Relation

Theorem: Every Cauchy sequence of real numbers has a convergent subsequence.

Theorem (Cauchy’s General Principle for Convergence): A sequence of real number is convergent if and only if it is a Cauchy sequence.

Theorem (nested intervals): Suppose that $\{I_n\}$ is a sequence of the closed interval such that ${I_n} = \left[a_n,b_n\right]$, $I_{n + 1}\subset I_n \,\,\,\forall \,\,\,n\ge 1$ and $(b_n-a_n)\to 0$ as $n\to \infty$ then $\cap {I_n}$ contains one and only one point.

Theorem (Bolzano-Weierstrass theorem): Every bounded sequence has a convergent subsequence.

Limit inferior of the sequence.

Limit superior of the sequence.

Theorem: If $\{s_n\}$ is a convergent sequence then $\lim_{n\to\infty} {s_n}=\lim_{n\to\infty} \left(\inf{s_n} \right)=\lim_{n\to\infty}\left(\sup {s_n}\right)$.

Infinite Series.

Theorem: If $\sum_{n=1}^\infty{a_n}$ converges then $\lim_{n\to\infty} {a_n}=0$.

Theorem (General Principle of Convergence): A series $\sum{a_n}$ is convergent if and only if for any real number $\varepsilon >0$, there exists a positive integer $n_0$ such that $\left|\sum_{i=m+1}^\infty {a_i}\right|< \varepsilon$ for all $n>m>n_0$.

Theorem: Let $\sum {a_n}$ be an infinite series of non-negative terms and let $\{s_n\}$ be a sequence of its partial sums then $\sum{a_n}$ is convergent if $\{s_n\}$ is bounded and it diverges if $\{s_n\}$ is unbounded.

Theorem (Comparison Test)

Theorem: Let $a_n>0, b_n>0$ and $\lim_{n\to\infty } \frac{a_n}{b_n}=\lambda\ne 0$ then the series $\sum{a_n}$ and $\sum{b_n}$ behave alike.

Theorem ( Cauchy Condensation Test )

Alternating Series.

Theorem (Alternating Series Test or Leibniz Test)

Absolute Convergence

Theorem: An absolutely convergent series is convergent.

Theorem (The Root Test)

Theorem (Ratio Test)

Theorem (Dirichlet): Suppose that $\{s_n\}, {s_n}={a_1}+{a_2}+{a_3}+...+ {a_n}$ is bounded. Let $\{b_n\}$ be positive term decreasing sequence such that $\lim_{n\to\infty}{b_n}=0$. Then $\sum{a_n b_n}$ is convergent.

Theorem: Suppose that $\sum{a_n}$ is convergent and that $\{b_n\}$ is monotonic convergent sequence then $\sum{a_n b_n}$ is also convergent.

Lot of examples.