Theory

Numerical series

What it means to add infinitely many terms, which tests establish the behavior of a series, and how a power series is studied.

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Review of sequences
Series, partial sums and behavior

Review of sequences

Two results on sequences that are used throughout the chapter.

1. Limit of a sequence

A sequence is a function $a : N \to R$ ; its value at $n$ is written $a_{n}$ . One writes $n \to + \infty lim a_{n} = L$ , with $L \in R$ , if for every $ε > 0$ there exists $N$ such that $∣ a_{n} - L ∣ < ε$ for every $n > N$ ; one writes $n \to + \infty lim a_{n} = + \infty$ if for every $M > 0$ there exists $N$ such that $a_{n} > M$ for every $n > N$ , and similarly for $- \infty$ . A sequence with a finite limit is called convergent, one with an infinite limit divergent, one without a limit indeterminate. The theorems of the chapter on limits hold for limits of sequences: uniqueness, sign permanence, comparison, algebra of limits, standard limits and the hierarchy of infinities.

$a_{n} = \frac{1}{n}$ converges to $0$ .
$a_{n} = n^{2}$ diverges to $+ \infty$ .
$a_{n} = (- 1)^{n}$ is indeterminate, because its values oscillate between $- 1$ and $1$ .

2. Bounded monotone sequences

An increasing sequence that is bounded above converges, and its limit is the supremum of its values; an increasing sequence that is unbounded above diverges to $+ \infty$ . The symmetric statements hold for decreasing sequences. Therefore a monotone sequence always has a limit, finite or infinite. The proof uses the properties of the supremum and is omitted. The theorem underpins the entire section on series with nonnegative terms, because their partial sums are increasing.

Series, partial sums and behavior

Series, convergence, sum

Given a sequence $(a_{n})$ , the series with general term $a_{n}$ is the sequence of partial sums

s_{N} = n = 1 \sum N a_{n} = a_{1} + a_{2} + \dots + a_{N} .

Depending on the limit of the partial sums, the series $n = 1 \sum \infty a_{n}$

converges, with sum $s \in R$ , if $s_{N} \to s$ ; in this case one writes $n = 1 \sum \infty a_{n} = s$ ;
diverges to $+ \infty$ or to $- \infty$ , if $s_{N} \to + \infty$ or $s_{N} \to - \infty$ ;
is indeterminate, if $s_{N}$ has no limit.

Studying the behavior of a series means establishing which of the three cases it falls into. For example $n = 1 \sum \infty (- 1)^{n}$ has partial sums $- 1, 0, - 1, 0, \dots$ , which have no limit, therefore the series is indeterminate.

Remark

The behavior of a series does not depend on its first terms: if a finite number of terms is modified, from some index on the partial sums change by a constant, therefore convergence, divergence and indeterminacy are preserved, while the sum in general changes. For this reason the hypotheses of the tests are required eventually, that is, from some index on, and the starting index of the series is chosen where the general term is defined.

The theory continues with 6 more sections (reference series, the necessary condition, series with nonnegative terms…), plus all worked examples, the solutions and the PDF.

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