Abel’s Theorem (one standard form): Let (an)(a_n)(an​) be a real or complex sequence such that the series ∑n=0∞an\sum_{n=0}^\infty a_n∑n=0∞​an​ converges to sss. For 0≤r<10\le r<10≤r<1, define f(r)=∑n=0∞anrn, f(r)=\sum_{n=0}^\infty a_n r^n, f(r)=∑n=0∞​an​rn, which converges for ∣r∣<1|r|<1∣r∣<1. Then lim⁡r→1−f(r)=s. \lim_{r\to 1^-} f(r)=s. limr→1−​f(r)=s. ...

Absolute convergence implies convergence: Let (an)(a_n)(an​) be a real or complex sequence. If the series ∑n=1∞∣an∣\sum_{n=1}^\infty |a_n|∑n=1∞​∣an​∣ converges, then the series ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ converges. ...

Corollary: Let (an)(a_n)(an​) be a real or complex sequence. If ∑n=1∞∣an∣\sum_{n=1}^\infty |a_n|∑n=1∞​∣an​∣ converges , then for every ε>0\varepsilon>0ε>0 there exists NNN such that for all m>n≥Nm>n\ge Nm>n≥N, ∑k=n+1m∣ak∣<ε. \sum_{k=n+1}^{m} |a_k| < \varepsilon. ∑k=n+1m​∣ak​∣<ε. Consequently, the partial sums of ∑an\sum a_n∑an​ form a Cauchy sequence , so ∑an\sum a_n∑an​ converges. ...

The absolute value of a real number x∈Rx\in\mathbb{R}x∈R is ∣x∣:={x,x≥0,−x,x<0. |x| := \begin{cases} x,& x\ge 0,\\ -x,& x<0. \end{cases} ∣x∣:={x,−x,​x≥0,x<0.​Equivalently, ∣x∣=x2|x|=\sqrt{x^2}∣x∣=x2​ and ∣x∣|x|∣x∣ is the distance from xxx to 000 in R\mathbb{R}R with the usual metric d(x,y)=∣x−y∣d(x,y)=|x-y|d(x,y)=∣x−y∣. Absolute value is fundamental for defining convergence, continuity, and error bounds in one-variable analysis. ...

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R be Riemann integrable . Proposition: The function ∣f∣|f|∣f∣ is Riemann integrable on [a,b][a,b][a,b]. Moreover, ∣∫abf(x) dx∣≤∫ab∣f(x)∣ dx. \left|\int_a^b f(x)\,dx\right|\le \int_a^b |f(x)|\,dx. ​∫ab​f(x)dx​≤∫ab​∣f(x)∣dx. ...

A series ∑n=1∞an\sum_{n=1}^\infty a_n∑n=1∞​an​ with an∈Ra_n\in\mathbb{R}an​∈R or C\mathbb{C}C is absolutely convergent if the series of absolute values converges : ∑n=1∞∣an∣ converges.\sum_{n=1}^\infty |a_n| \ \text{converges}.n=1∑∞​∣an​∣ converges.Absolute convergence is stronger than ordinary convergence and has powerful consequences: in R\mathbb{R}R or C\mathbb{C}C it implies convergence of ∑an\sum a_n∑an​ and stability under rearrangement . ...

Additivity and linearity (Riemann integral): Let f,g:[a,b]→Rf,g:[a,b]\to\mathbb{R}f,g:[a,b]→R be Riemann integrable and let α,β∈R\alpha,\beta\in\mathbb{R}α,β∈R. Then: αf+βg\alpha f+\beta gαf+βg is Riemann integrable on [a,b][a,b][a,b], and $ \int_a^b (\alpha f(x)+\beta g(x)),dx \alpha\int_a^b f(x),dx+\beta\int_a^b g(x),dx. $ For any c∈[a,b]c\in[a,b]c∈[a,b], fff is Riemann integrable on [a,c][a,c][a,c] and on [c,b][c,b][c,b], and ∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx. \int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx. ∫ab​f(x)dx=∫ac​f(x)dx+∫cb​f(x)dx. Additivity and linearity (Riemann–Stieltjes integral): Let γ:[a,b]→R\gamma:[a,b]\to\mathbb{R}γ:[a,b]→R be increasing . If f,gf,gf,g are Riemann–Stieltjes integrable with respect to γ\gammaγ on [a,b][a,b][a,b] and α,β∈R\alpha,\beta\in\mathbb{R}α,β∈R, then αf+βg\alpha f+\beta gαf+βg is Riemann–Stieltjes integrable with respect to γ\gammaγ and $ \int_a^b (\alpha f+\beta g),d\gamma \alpha\int_a^b f,d\gamma+\beta\int_a^b g,d\gamma. Moreover,forany Moreover, for any Moreover,foranyc\in[a,b],, , \int_a^b f,d\gamma=\int_a^c f,d\gamma+\int_c^b f,d\gamma. $ ...

Algebra of limits for sequences: Let (an)(a_n)(an​) and (bn)(b_n)(bn​) be sequences in R\mathbb{R}R (or C\mathbb{C}C) with an→aa_n\to aan​→a and bn→bb_n\to bbn​→b. Then: an+bn→a+ba_n+b_n \to a+ban​+bn​→a+b, anbn→aba_n b_n \to aban​bn​→ab, for any scalar ccc, can→cac a_n \to cacan​→ca, if b≠0b\ne 0b=0 and bn≠0b_n\ne 0bn​=0 eventually, then an/bn→a/ba_n/b_n \to a/ban​/bn​→a/b, in C\mathbb{C}C, an‾→a‾\overline{a_n}\to \overline{a}an​​→a and ∣an∣→∣a∣|a_n|\to |a|∣an​∣→∣a∣. These rules make limits computationally usable and are proved directly from the ε\varepsilonε–NNN definition (often together with basic inequalities). ...

Let f,g:[a,b]→Rf,g:[a,b]\to\mathbb{R}f,g:[a,b]→R be Riemann integrable . Proposition: For any α,β∈R\alpha,\beta\in\mathbb{R}α,β∈R, the function αf+βg\alpha f+\beta gαf+βg is Riemann integrable. The product fgfgfg is Riemann integrable. Consequently, f2f^2f2 is Riemann integrable. These closure properties are essential: they guarantee the Riemann integral behaves well under the usual algebraic operations on functions. Proof sketch: Linearity is standard from linearity of sums and the integrability definition via upper /lower sums . For products, note that Riemann integrable functions are bounded , so ∣f∣≤M|f|\le M∣f∣≤M and ∣g∣≤N|g|\le N∣g∣≤N. Use the identity fg=14((f+g)2−(f−g)2) fg=\frac{1}{4}\bigl((f+g)^2-(f-g)^2\bigr) fg=41​((f+g)2−(f−g)2) to reduce product integrability to integrability of squares, and show that if hhh is integrable then h2h^2h2 is integrable (e.g., by continuity of x↦x2x\mapsto x^2x↦x2 and the Lebesgue criterion , or by oscillation estimates). ...