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_...
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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...
Continuity of the Derivative
Darboux Theorem $f:[a,b]\to\mathbb{R}$ is differentiable on $[a,b]$. Assume that $f^\prime(a)<f^\prime(b)$. Then $\forall\lambda \in (f^\prime(a), f^\prim...
Intermediate Value Theorem for derivatives
Theorem 4.2.13 (Intermediate Value Theorem for derivative) Let $f:I\to\mathbb{R}$ be differentiable function on the interval $I=[a,b]$. Then given $a,b\in I$...