From 0a6b8f618ab68f139c604314710aa9b921c9df68 Mon Sep 17 00:00:00 2001 From: arc Date: Tue, 6 Jan 2026 11:56:59 -0700 Subject: [PATCH] vault backup: 2026-01-06 11:56:59 --- .../Dividing Polynomials.md | 0 education/math/{MATH1050 => MATH1050 (algebra)}/Domain.md | 0 .../Exponents/Exponents.md | 0 .../Exponents/Logarithms.md | 0 .../Exponents/Quadratics.md | 0 .../{MATH1050 => MATH1050 (algebra)}/Inverse Functions.md | 0 .../math/{MATH1050 => MATH1050 (algebra)}/Matrices.md | 0 .../{MATH1050 => MATH1050 (algebra)}/Partial Fractions.md | 0 .../Polynomial Fractions.md | 0 .../Rational Inequalities.md | 0 .../Standard forms of circles.md | 0 .../Systems of Equations.md | 0 .../{Sequences.md => Sequences and Series.md} | 7 ++++++- 13 files changed, 6 insertions(+), 1 deletion(-) rename education/math/{MATH1050 => MATH1050 (algebra)}/Dividing Polynomials.md (100%) rename education/math/{MATH1050 => MATH1050 (algebra)}/Domain.md (100%) rename education/math/{MATH1050 => MATH1050 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Polynomials.md similarity index 100% rename from education/math/MATH1050/Dividing Polynomials.md rename to education/math/MATH1050 (algebra)/Dividing Polynomials.md diff --git a/education/math/MATH1050/Domain.md b/education/math/MATH1050 (algebra)/Domain.md similarity index 100% rename from education/math/MATH1050/Domain.md rename to education/math/MATH1050 (algebra)/Domain.md diff --git a/education/math/MATH1050/Exponents/Exponents.md b/education/math/MATH1050 (algebra)/Exponents/Exponents.md similarity index 100% rename from education/math/MATH1050/Exponents/Exponents.md rename to education/math/MATH1050 (algebra)/Exponents/Exponents.md diff --git a/education/math/MATH1050/Exponents/Logarithms.md b/education/math/MATH1050 (algebra)/Exponents/Logarithms.md similarity index 100% rename from education/math/MATH1050/Exponents/Logarithms.md rename to education/math/MATH1050 (algebra)/Exponents/Logarithms.md diff --git a/education/math/MATH1050/Exponents/Quadratics.md b/education/math/MATH1050 (algebra)/Exponents/Quadratics.md similarity index 100% rename from education/math/MATH1050/Exponents/Quadratics.md rename to education/math/MATH1050 (algebra)/Exponents/Quadratics.md diff --git a/education/math/MATH1050/Inverse Functions.md b/education/math/MATH1050 (algebra)/Inverse Functions.md similarity index 100% rename from education/math/MATH1050/Inverse Functions.md rename to education/math/MATH1050 (algebra)/Inverse Functions.md diff --git a/education/math/MATH1050/Matrices.md b/education/math/MATH1050 (algebra)/Matrices.md similarity index 100% rename from education/math/MATH1050/Matrices.md rename to education/math/MATH1050 (algebra)/Matrices.md diff --git a/education/math/MATH1050/Partial Fractions.md b/education/math/MATH1050 (algebra)/Partial Fractions.md similarity index 100% rename from education/math/MATH1050/Partial Fractions.md rename to education/math/MATH1050 (algebra)/Partial Fractions.md diff --git a/education/math/MATH1050/Polynomial Fractions.md b/education/math/MATH1050 (algebra)/Polynomial Fractions.md similarity index 100% rename from education/math/MATH1050/Polynomial Fractions.md rename to education/math/MATH1050 (algebra)/Polynomial Fractions.md diff --git a/education/math/MATH1050/Rational Inequalities.md b/education/math/MATH1050 (algebra)/Rational Inequalities.md similarity index 100% rename from education/math/MATH1050/Rational Inequalities.md rename to education/math/MATH1050 (algebra)/Rational Inequalities.md diff --git a/education/math/MATH1050/Standard forms of circles.md b/education/math/MATH1050 (algebra)/Standard forms of circles.md similarity index 100% rename from education/math/MATH1050/Standard forms of circles.md rename to education/math/MATH1050 (algebra)/Standard forms of circles.md diff --git a/education/math/MATH1050/Systems of Equations.md b/education/math/MATH1050 (algebra)/Systems of Equations.md similarity index 100% rename from education/math/MATH1050/Systems of Equations.md rename to education/math/MATH1050 (algebra)/Systems of Equations.md diff --git a/education/math/MATH1220 (calc II)/Sequences.md b/education/math/MATH1220 (calc II)/Sequences and Series.md similarity index 95% rename from education/math/MATH1220 (calc II)/Sequences.md rename to education/math/MATH1220 (calc II)/Sequences and Series.md index cd17f06..cdcb0b7 100644 --- a/education/math/MATH1220 (calc II)/Sequences.md +++ b/education/math/MATH1220 (calc II)/Sequences and Series.md @@ -147,4 +147,9 @@ The behavior a given power series falls into one of three cases: $$ \frac{x}{3}$$ 2. For a geometric series, it converges when the ratio $r$ is less than one. Written as an equality, this gives us: $$ |r| < 1 \to |\frac{x}{3}| < 1 \to |x| < 3 $$ -This means that the interval of convergence is $(-3, 3)$, given it diverges at both interval endpoints. \ No newline at end of file +This means that the interval of convergence is $(-3, 3)$, given it diverges at both interval endpoints. + +## Integral test for series +The integral test determines if an infinite series converges or diverges by comparing it to an improper integral. If the integral diverges ($= \infty$), then the series also diverges. If the integral converges, then the series is + +Given the series $\sum_{n=0}^\infty a_n$, the improper integral would be of the below form: