Pointwise Convergence and Uniform Convergence
Definition 8.1.1 (Pointwise Convergence) A sequence of real valued functions $\{f_n\}$ defined on a set $E (\subset \mathbb{R})$ converges pointwise on $E$ i...
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Dirichlet Test and Absolute Convergence
Theorem 7.2.1 Let $\{a_k\}$ and $\{b_k\}$ be a sequence of real numbers. Set $A_0:=0, A_n:=\sum_{k=1}^n a_k$ for $n\geq 1$. Then if $1\leq p \leq q$,
Convergence Test
Definition Let $\{a_n\}$ be a sequence. Define a new sequence of $s_n := \sum_{k=1}^na_k$ for $n\geq 1$. We call the $s_n$ is the $n$-th partial sum of $\{a_...
Properties of the Riemann Integral
Theorem 6.1.9 Let $f:[a,b]\to\mathbb{R}$ be a bounded real valued Riemann integrable function with $Range f \subset [c,d]$. Let $\varphi:[c,d]\to\mathbb{R}$ ...
Riemann Integral
Definition (upper sum, lower sum) Let $[a,b]$ with $a<b$ be a closed and bounded interval in $\mathbb{R}$. By a partition of $\mathscr{P}$ of $[a,b]$ we m...