Infinite series
Just write definitions and theorems
11.1 Preliminaries
Definition 11.1.1: Let $\{a_j \}$ be a sequence of real numbers.
i) For each $k$ in $\mathbb{N}$, define $s_k=\sum_{j=1}^k a_j$. The squence $\{s_k\}$ is called the sequence of $k$th partial sums.
ii) An infinite series, denoted $\sum_{j=1}^\infty a_j$($=\lim_{k\rightarrow \infty} \sum_{j=1}^\infty a_j$), is the sequence $\{s_k\}$ of partial sums of the numbers $a_1, a_2, \cdots, a_k, \cdots$
Definition 11.1.2: Given an infinite series $\sum_{j=1}^\infty a_j$, let $\{s_k\}$ denote its sequence of $k$th partial sums.
i) If the sequence $\{s_k\}$ converges (to $S$), then the infinite series $\sum_{j=1}^\infty a_j$ is said to converges (to $S$) and we writes $$\sum_{j=1}^\infty a_j = S$$ $S$ is called the sum of the series.
ii) If the sequence $\{s_k\}$ diverges, then the infinite series $\sum_{j=1}^\infty a_j$ is said to diverges.
Theorem 11.1.1: A series $\sum_{j=1}^\infty a_j$ converges if and only if the tail of the series is eventually arbitraily small. That is, for each $\epsilon > 0$, there exists a $k_0$ such that, for $k\ge k_0$, $$\left| \sum_{j=k+1}^\infty a_j \right| < \epsilon$$
Theorem 11.1.2: An infinite series $\sum_{j=1}^\infty a_j$ converges if and only if the sequence $\{s_k \}$ of $k$th partial sums is Cauchy.
Proof: $\mathbb{R}$ is complete.
Corollary 11.1.3: If the infinite seires $\sum_{j=1}^\infty a_j$ converges, then $\lim_{j\rightarrow \infty} a_j = 0$.
11.2 Convergence Tests (Positive Series)
Theorem 11.2.1 The Comparison Test: Let $\sum_{j=1}^\infty a_j$ and $\sum_{j=1}^\infty b_j$ be two series of positive terms such that, for some $j_0$, $$0 < a_j \le b_j,\ \mbox{for all } j \ge j_0$$
i) If $\sum_{j=1}^\infty b_j$ converges, then $\sum_{j=1}^\infty a_j$ also converges.
ii) If $\sum_{j=1}^\infty a_j$ diverges, then $\sum_{j=1}^\infty b_j$ also diverges.
Theorem 11.2.2 The Limit Comaparison Test: Suppose that $\sum_{j=1}^\infty a_j$ and $\sum_{j=1}^\infty b_j$ are two series of positive terms such that $\lim_{j\rightarrow \infty} a_j/b_j=L$ exists and is positive. Then $\sum_{j=1}^\infty a_j$ converges if and only if $\sum_{j=1}^\infty b_j$ converges.
Theorem 11.2.3 The Integral Test: Suppose that $\sum_{j=1}^\infty a_j$ is a series of positive terms and that $\{ a_j\}$ is a decreasing sequence converging to $0$. Suppose that $f$ is a continuous, monotone decreasing function on $[1,\infty)$ such that $f(j)=a_j$, for $j=1,2,3,\cdots$. The series $\sum_{j=1}^\infty a_j$ converges if and only is $\lim_{k\rightarrow \infty} \int_1^k f(x) dx$ exists.
Theorem 11.2.4 d'Alembert's Ratio Test: Let $\sum_{j=1}^\infty a_j$ be a seires of positive terms such that $$\lim_{j\rightarrow \infty} \frac{a_{j+1}}{a_j} = L$$ exists.
i) If $L<1$, then $\sum_{j=1}^\infty a_j$ converges.
ii) If $L>1$, then $\sum_{j=1}^\infty a_j$ diverges.
iii) If $L=1$, no information is gained form the test.
Theorem 11.2.5 Cauchy's Root Test: Let $\sum_{j=1}^\infty a_j$ be a seires of positive terms such that $\lim_{j\rightarrow \infty} (a_j)^{1/j} = L$ exists.
i) If $L<1$, then $\sum_{j=1}^\infty a_j$ converges.
ii) If $L>1$, then $\sum_{j=1}^\infty a_j$ diverges.
iii) If $L=1$, no information is gained from the test.
Theorem 11.2.6 The Ratio Test, Form II: Let $\sum_{j=1}^\infty a_j$ be a series of positive terms.
i) If $\lim\sup a_{j+1}/a_j < 1$, then $\sum_{j=1}^\infty a_j$ converges.
ii) If $\lim\inf a_{j+1}/a_j >1$, then $\sum_{j=1}^\infty a_j$ diverges.
iii) If $\lim \inf a_{j+1}/a_j \le 1 \le \lim\sup a_{j+1}/a_j$, then the test provides no information.
Theorem 11.2.7 The Root Test, Form II: Let $\sum_{j=1}^\infty a_j$ be a series of positive terms.
i) If $\lim \sup (a_j)^{1/j}<1$, then $\sum_{j=1}^\infty a_j$ converges.
ii) If $\lim \sup (a_j)^{1/j} >1$, then $\sum_{j=1}^\infty a_j$ diverges.
iii) If $\lim \sup (a_j)^{1/j} =1$, then the test provides no information.
11.3 Absolute Convergence
Theorem 11.3.1: If a series $\sum_{j=1}^\infty a_j$ converges absolutely, then is converges.
Theorem 11.3.2: If $\sum_{j=1}^\infty a_j$ converges absolutely, then the two series $\sum_{j=1}^\infty a^+_j$ and $\sum_{j=1}^\infty a^-_j$ also converges absolutely. Further, $\sum_{j=1}^\infty a_j = \sum_{j=1}^\infty a_j^+ - \sum_{j=1}^\infty a_j^-$ and $\sum_{j=1}^\infty \left| a_j \right| = \sum_{j=1}^\infty a^+_j + \sum_{j=1}^\infty a_j^-$
11.4 Conditional Convergence
Theorem 11.4.1 The Alternating Seires Test: If $\{ a_j\}$ is a monotone decreasing sequence of positive numbers suth that $\lim_{j\rightarrow \infty} a_j = 0$, then $\sum_{j=1}^\infty (-1)^{j+1}a_j$ converges.
Theorem 11.4.2 Abel's Partial Summation Formula: Given a seires of the form $\sum_{j=1}^\infty a_jb_j$, let $s_k = \sum_{j=1}^\k a_j$. Then $$\sum_{j=1}^k a_j b_j = s_k b_{k+1} - \sum_{j=1}^k s_j(b_{j+1} - b_j)
Theorem 11.4.3 Abel's Test: Suppose that $\sum_{j=1}^\infty a_j$ is a convergent series and that $\{b_j\}$ is a monotone, convergent sequence. Then the series $\sum_{j=1}^\infty a_jb_j$ converges.
Theorem 11.4.4 Dirichlet's Test: Let $\sum_{j=1}^\infty a_jb_j$ be a series of real numbers. If the sequence $\{s_k\}$ of $k$th partial sums of the series $\sum_{j=1}^\infty a_j$ is bounded and if $\{b_j\}$ is a sequence of positive numbers that converges monotonically 0, then the series $\sum_{j=1}^\infty a_jb_j$ converges.
Theorem 11.4.5: For any real number $x$ that is not an even multiple of $\pi$, we have
i) $$ \sum_{j=1}^k \cos(jx) = \frac{\cos [(k+1)x/2] \sin (kx/2)}{\sin (x/2)}$$
ii) $$\sum_{j=1}^k \sin(jx) = \frac{\sin [(k+1)x/2]\sin(kx/2)}{\sin(x/2)}$$
Theorem 11.4.6: If $\sum_{j=1}^\infty a_j$ converges conditionally, then the two series $\sum_{j=1}^\infty a_j^+$ and $\sum_{j=1}^\infty a_j^-$ diverge to infinity.
Theorem 11.4.7 Weierstrass: Let $\sum_{j=1}^\infty a_j$ be a conditionally convergent series.
i) Let $L$ be any real number. Then there exists a rearrangement $\sum_{j=1}^\infty a_{\phi(j)}$ of $\sum_{j=1}^\infty a_j$ that converges to $L$.
ii) If $L_1$ and $L_2$ are any two real numbers or $\pm \infty$ with $L_1 < L_2$, then there exists a rearrangement $\sum_{j=1}^\infty a_{\phi(j)}$ of $\sum_{j=1}^\infty a_j$ such that $\lim \inf t_k = L_1$ and $\lim \sup t_k = L_2$, where $t_k=\sum_{j=1}^\infty a_{\phi(j)}$
Theorem 11.4.8: If $\sum_{j=1}^\infty a_j$ converges absolutely to $S$, then any rearrangement $\sum_{j=1}^\infty a_{\phi(j)}$ of $\sum_{j=1}^\infty a_j$ also converges to $S$.
11.5 The Cauchy Product
Definition 11.5.1 The Cauchy Product: The Cauchy product of two series $\sum_{j=0}^\infty a_j$ and $\sum_{j=0}^\infty b_j$ is defined to be $$(\sum_{j=0}^\infty a_j)(\sum_{j=0}^\infty b_j) = \sum_{k=0}^\infty c_k$$ where $c_k = \sum_{j=0}^k a_jb_{k-j}$
Theorem 11.5.1: The Cauchy product of two absolutely convergent seires $\sum_{j=0}^\infty a_j$ and $\sum_{j=0}^\infty b_j$ converges absolutely to $(\sum_{j=0}^\infty a_j)(\sum_{j=0}^\infty b_j)$
11.6 Cesaro Summability
Definition 11.6.1: If the sequence $\{\sigma_k\}$ converges to $\sigma$, then we say that the series $\sum_{j=1}^\infty a_j$ is Cesaro summable and converges $(C,1)$ to $\sigma$. We write $$\sum_{j=1}^\infty a_j = \sigma (C,1)$$
Theorem 11.6.1: If $\sum_{j=1}^\infty a_j$ converges to $S$, then $\sum_{j=1}^\infty a_j$ is Cesaro summable to $S$.
Reference
Douglass - Introduction to Mathematical Analysis