Sequences and Series

The second page of the analysis spine. With completeness of $\mathbb{R}$ in hand (real-numbers.md), we develop the theory of limits of sequences — the prototype of every limiting process in calculus — and of infinite series, including the uniform convergence needed to differentiate and integrate limits term by term.

References: Rudin, Principles of Mathematical Analysis, ch. 3, 7; Tao, Analysis I.

1. Convergence of Sequences

A sequence in $\mathbb{R}$ is a function $n\mapsto a_n$, $n\in \mathbb{N}$. It converges to $L\in \mathbb{R}$, written $a_n\to L$ or ${\lim }_n{a}_n=L$, if

$$\forall \epsilon >0\exists N\in \mathbb{N}\forall n\ge N:|a_n-L|<\epsilon .$$

A sequence is bounded if $\{a_n\}$ is a bounded set. Convergent sequences are bounded, and limits are unique. The algebra of limits holds: if $a_n\to A$ and $b_n\to B$ then $a_n+b_n\to A+B$, $a_n{b}_n\to AB$, and $a_n/b_n\to A/B$ when $B\ne 0$. The squeeze theorem: if $a_n\le c_n\le b_n$ and $a_n,b_n\to L$ then $c_n\to L$.

2. Monotone Sequences and Completeness

A sequence is monotone increasing if $a_n\le a_{n+1}$ for all $n$ (decreasing dually).

Monotone Convergence Theorem. A monotone increasing sequence converges iff it is bounded above, and then ${\lim }_n{a}_n={\sup }_n{a}_n$.

Proof. If bounded above, $L={\sup }_n{a}_n$ exists by completeness (real-numbers.md §2). Given $\epsilon >0$, some $a_N>L-\epsilon$; monotonicity gives $L-\epsilon <a_n\le L$ for all $n\ge N$. $\square$

This is the first concrete payoff of completeness, and the basis for defining quantities like $e=\lim (1+1/n{)}^n$.

3. Subsequences and Bolzano–Weierstrass

A subsequence $(a_{n_k})$ is obtained by choosing indices $n_1<n_2<\cdots$. A point $L$ is a subsequential limit (or cluster point) if some subsequence converges to it.

Bolzano–Weierstrass Theorem. Every bounded sequence in $\mathbb{R}$ has a convergent subsequence.

Proof. Bisection: enclose the sequence in $[a,b]$, repeatedly halve, keeping a half containing infinitely many terms. The nested intervals (real-numbers.md §4.4) shrink to a point $L$; pick one term from each interval to build a subsequence converging to $L$. $\square$

This compactness statement is the engine behind the extreme value theorem (continuity.md) and generalizes to ${\mathbb{R}}^n$ and metric spaces in metric-spaces.md.

4. Cauchy Sequences and Completeness Restated

A sequence is Cauchy if

$$\forall \epsilon >0\exists N\forall m,n\ge N:|a_m-a_n|<\epsilon .$$

Cauchy Criterion. A sequence in $\mathbb{R}$ converges iff it is Cauchy.

Proof. ($\Rightarrow$) immediate from the triangle inequality. ($\Leftarrow$) A Cauchy sequence is bounded, so by Bolzano–Weierstrass has a convergent subsequence $a_{n_k}\to L$; the Cauchy condition then forces the whole sequence to $L$. $\square$

The value of the Cauchy criterion is that it tests convergence without knowing the limit in advance — essential for series and for completeness of function spaces. "Complete" as a metric-space property (metric-spaces.md) means exactly "every Cauchy sequence converges."

5. limsup and liminf

For a bounded sequence define

$$\underset{n}{\mathrm{limsup}}{a}_n=\underset{N\to \infty }{\lim }\underset{n\ge N}{\sup }{a}_n,\underset{n}{\mathrm{liminf}}{a}_n=\underset{N\to \infty }{\lim }\underset{n\ge N}{\inf }{a}_n.$$

Both always exist in $\overline{\mathbb{R}}$ (the inner sup/inf are monotone in $N$). They are the largest and smallest subsequential limits, and $a_n\to L⟺\mathrm{limsup}{a}_n=\mathrm{liminf}{a}_n=L$. These give convergence tests (below) that require no candidate limit.

6. Infinite Series

Given $(a_n)$, the series ${\sum }_{n=1}^{\infty }{a}_n$ is the sequence of partial sums $s_N={\sum }_{n=1}^N{a}_n$; it converges to $s$ if $s_N\to s$. A necessary condition is $a_n\to 0$ (not sufficient: the harmonic series $\sum 1/n$ diverges).

Cauchy criterion for series. $\sum a_n$ converges iff for all $\epsilon >0$ there is $N$ with $\left|{\sum }_{n=m}^k{a}_n\right|<\epsilon$ for all $k\ge m\ge N$.

6.1 Absolute vs conditional convergence

$\sum a_n$ converges absolutely if $\sum |a_n|$ converges. Absolute convergence implies convergence (by the Cauchy criterion and the triangle inequality). A series that converges but not absolutely (e.g. the alternating harmonic series $\sum (-1{)}^{n+1}/n$) converges conditionally. Absolutely convergent series may be reordered freely; conditionally convergent ones may not (Riemann rearrangement theorem: their terms can be rearranged to sum to any value).

6.2 Convergence tests

• Comparison: $0\le a_n\le b_n$ and $\sum b_n<\infty \Rightarrow \sum a_n<\infty$.
• Ratio: if $\mathrm{limsup}|a_{n+1}/a_n|<1$, converges absolutely; if $>1$, diverges.
• Root: with $\rho =\mathrm{limsup}|a_n{|}^{1/n}$: $\rho <1$ converges, $\rho >1$ diverges (sharper than the ratio test).
• Integral test: for positive decreasing $f$, $\sum f(n)$ and ${\int }_1^{\infty }f$ converge together (links to riemann-integral.md).
• Alternating series test: if $b_n\downarrow 0$, then $\sum (-1{)}^n{b}_n$ converges.

7. Sequences and Series of Functions

Let $f_n:E\to \mathbb{R}$. Pointwise convergence $f_n\to f$ means $f_n(x)\to f(x)$ for each fixed $x$. This is too weak to preserve continuity or commute with limits/integrals. The right notion is:

Uniform convergence. $f_n\to f$ uniformly on $E$ if $$\underset{x\in E}{\sup }|f_n(x)-f(x)|\to 0,$$ i.e. one $N$ works for all $x$ simultaneously.

Weierstrass $M$-test. If $|f_n(x)|\le M_n$ for all $x$ and $\sum M_n<\infty$, then $\sum f_n$ converges uniformly.

The three theorems that make uniform convergence the "correct" notion:

• Continuity is preserved: a uniform limit of continuous functions is continuous (continuity.md).
• Integration commutes with the limit: if $f_n\to f$ uniformly on $[a,b]$ and each $f_n$ is integrable, then ${\int }_a^b{f}_n\to {\int }_a^{b}f$ (riemann-integral.md).
• Differentiation (with a hypothesis on the derivatives): if $f_n\to f$ pointwise and $f_n^{\prime }\to g$ uniformly, then $f$ is differentiable and $f^{\prime }=g$ (differentiation.md).

7.1 Power series

A power series ${\sum }_{n=0}^{\infty }{c}_n(x-a{)}^n$ has a radius of convergence $R=1/\mathrm{limsup}|c_n{|}^{1/n}$ (Cauchy–Hadamard): it converges absolutely for $|x-a|<R$ and diverges for $|x-a|>R$. Convergence is uniform on every compact subinterval of $(a-R,a+R)$, so power series may be differentiated and integrated term by term inside their interval of convergence. This is what makes analytic functions (differentiation.md §6) so well-behaved.

8. Where this page is used

• Cauchy criterion / completeness generalize to ${\mathbb{R}}^n$ and abstract metric spaces in metric-spaces.md, powering the contraction-mapping theorem.
• Bolzano–Weierstrass underlies compactness and the extreme value theorem in continuity.md.
• Uniform convergence licenses the term-by-term operations used throughout differentiation.md and riemann-integral.md.

Next: Limits and Continuity.