Calculus Lecture Notes: Infinite Series and Convergence Criteria

Vocabulary flashcards covering the definitions, key convergence theorems, criteria, and classic examples of infinite series.

Series

The sum of the infinitely many terms of a sequence $(a_n)$, denoted as $\sum_{n=1}^{+\infty} a_n$ and formally defined as the sequence of its partial sums $(s_n)$.

Partial Sums

The sequence $(s_n)$ formed by summing a finite number of terms of a sequence $(a_n)$, where $s_1 = a_1$ and $s_n = s_{n-1} + a_n = a_1 + a_2 + \dots + a_n$.

Convergent Series

A series $\sum_{n=1}^{+\infty} a_n$ whose sequence of partial sums $(s_n)$ has a finite limit $S$ as $n \to +\infty$, written as $\sum_{n=1}^{+\infty} a_n = S$.

Irregular Series

A series $\sum_{n=1}^{+\infty} a_n$ for which the sequence of partial sums $(s_n)$ is irregular, meaning $\lim_{n \to +\infty} s_n$ does not exist.

Mengoli Series

The series given by $\sum_{n=1}^{+\infty} \frac{1}{n(n+1)} = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots$, which converges and has a sum equal to $1$.

Geometric Series

A series with ratio $q$ of the form $\sum_{n=0}^{+\infty} q^n = 1 + q + q^2 + q^3 + \dots$, whose partial sum equals $s_n = \frac{1-q^{n+1}}{1-q}$ for $q \neq 1$.

Convergence Behavior of Geometric Series

The geometric series $\sum_{n=0}^{+\infty} q^n$ converges to $\frac{1}{1-q}$ if $|q| < 1$, diverges to $+\infty$ if $q \ge 1$, and is irregular if $q \le -1$.

Harmonic Series

The series $\sum_{n=1}^{+\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \dots$, which is divergent to $+\infty$.

Necessary Condition for Convergence

Theorem 407 stating that if the series $\sum_{n=0}^{+\infty} a_n$ converges, then $\lim_{n \to +\infty} a_n = 0$; if $a_n$ does not converge to $0$, the series cannot converge.

Regularity Test for Positive Series

Proposition 409 stating that each series with positive terms ($a_n \ge 0$) is regular (either convergent or positively divergent), and it converges if and only if its sequence of partial sums is bounded from above.

Generalized Harmonic Series

The series $\sum_{n=1}^{+\infty} \frac{1}{n^\alpha}$ with $\alpha \in \mathbb{R}$, which converges for $\alpha > 1$ and diverges to $+\infty$ for $\alpha \le 1$.

Asymptotic Comparison Criterion

Proposition 417 stating that if $\sum_{n=0}^{+\infty} a_n$ and $\sum_{n=0}^{+\infty} b_n$ are two series with strictly positive terms and $a_n \sim b_n$, then both series have the same character (both converge or both diverge to $+\infty$).

Comparison Criterion

Proposition 410 stating that for two series with positive terms where $a_n \le b_n$ eventually: if $\sum a_n$ diverges positively, then $\sum b_n$ diverges positively; if $\sum b_n$ converges, then $\sum a_n$ converges.

Logarithmic Generalized Harmonic Series

The series $\sum_{n=2}^{+\infty} \frac{1}{n^\alpha \ln^\beta(n)}$, which converges for $\alpha > 1$ with any $\beta \in \mathbb{R}$ as well as for $\alpha = 1$ and $\beta > 1$, and diverges for $\alpha < 1$ with any $\beta \in \mathbb{R}$ as well as for $\alpha = 1$ and $\beta \le 1$.

Exponential Series

The power series defined by $\sum_{n=0}^{+\infty} \frac{x^n}{n!} = e^x$ for every $x \in \mathbb{R}$, which gives $\sum_{n=0}^{+\infty} \frac{1}{n!} = e$ when $x = 1$.

Absolute Convergence

Definition 429 stating that a series $\sum_{n=0}^{+\infty} a_n$ is absolutely convergent if the series of its absolute values, $\sum_{n=0}^{+\infty} |a_n|$, converges.

Absolute Convergence Theorem

Theorem 430 stating that if a series converges absolutely, then it converges.

Alternating Harmonic Series Sum

Proposition 435 stating that the alternating series $\sum_{n=1}^{+\infty} \frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \dots$ converges to $\ln(2)$.