S. D. V. Stephens

notes / Studies in Real and Complex Analysis / chapter 5: Analysis in One Variable

19 Sequences, Series and Power Series

· Denis Auroux

Original Professor Auroux’s handwritten notes (PDF), behind a password for now:

Real analysis!

References: Rudin, Principles of Mathematical Analysis (Chapter 4 on continuity as review; Chapter 3 on sequences and series in ; the beginnings of Chapters 7 and 8 on sequences and series of functions and power series; then the start of Chapter 5 on differentiation), and McMullen’s Math 55b notes (Section 3 as review, then Section 4 and the start of Section 5).

The basic object of real analysis is functions (or defined on a subset of , the domain of ), together with

  • their continuity, differentiability, integrals, …
  • sequences and series of functions.

Review: real functions

  • Continuity at : such that .

    More general limits: , , …; infinite limits and limits at infinity can be understood as taking place in the compactification , or explicitly, e.g. means such that .

  • Things we’ve already seen, using compactness and connectedness of :

    • continuous functions are uniformly continuous (the same works at all ). Continuity is ; uniform continuity is .
    • intermediate value theorem: is connected, so it contains all reals between and .
    • extreme value theorem: is compact, so it is bounded and contains its inf and sup.

  • Two topologies on spaces of functions so far (here , but similarly for ):

    • pointwise if , in ( product topology);
    • uniformly if ( uniform topology).

    We’ve seen: if the are continuous and uniformly, then is continuous (Lecture 4). Also, spaces of functions , () with the uniform topology are complete metric spaces, and (continuous functions) is a closed subspace, hence complete as well (but… unless we restrict to bounded functions, isn’t quite a metric).

    Analysis considers many different spaces of functions (bounded, integrable, continuous, differentiable) and topologies (often but not always metrics) on them.

  • Beyond polynomials and a few other explicit examples, many functions are defined using limits of sequences or series. A key example (also for complex analysis!):

Definition 19.1. Power series are expressions of the form for some coefficients .

(Writing this expression doesn’t in itself guarantee the series converges for any !) We’ll want to understand convergence (pointwise, uniformly over certain subsets of , …), so basic facts about real sequences and series in come in handy.

More review: sequences and series in

  • is complete, so a sequence in converges iff it is Cauchy.
  • Compactness of implies every bounded sequence in has convergent subsequences.
  • A monotonic sequence (e.g. ) converges iff it is bounded (and then ).
  • We sometimes write or ; this can be interpreted as convergence in the compactification . Such a sequence is still said to diverge.
  • If is bounded then is decreasing, and Similarly .

Example 19.2. and both have , .

Recall a series converges iff its partial sums form a convergent sequence, and then we write for the limit.

  • If converges then (by the Cauchy criterion for : ), but not vice versa ( diverges even though ).

  • For , converges iff the partial sums are bounded (since is increasing).

  • Hence (comparison criterion): if , then

  • The geometric series: if (it does not converge if ; in fact the terms !).

  • converges iff . The proof is a variant of the comparison argument: the block has terms, each between and , so a geometric series! Hence the partial sums are bounded iff . (My margin note beside that inequality: “where tf did this come from”.)

  • . (The equality comes from applying the binomial theorem to and showing that for fixed , increases with and as .)

    is irrational: denoting the partial sum , we have , so isn’t an integer multiple of for any , so it is not rational.

Definition 19.3. A series is absolutely convergent if converges.

If converges then converges (using Cauchy: ), but not vice versa, e.g. alternating series: if then converges! (Proof: the odd and even partial sums increase and decrease towards a common limit.)

Example 19.4.

Absolutely convergent series can be safely rearranged (), multiplied, etc.; others, not always.

Theorem 19.5 (Root test). If then converges (absolutely) (comparison with a geometric series). If then it diverges ().

This is used to great effect for power series!

Power series

Definition 19.6. The radius of convergence of is

Theorem 19.7.

  • converges pointwise for all such that ;
  • the convergence is uniform on for all (but not necessarily on );
  • thus is continuous over ;
  • the series diverges whenever ; at it may converge or diverge.

Proof.

  • Root test: , so the series converges for and diverges for .
  • Uniform convergence: if then The series converges by the root test, so , and which is uniform convergence.
  • The partial sums are continuous, so is continuous on by uniform convergence, hence continuous on .

∎

Example 19.8.

  • for (, so ). It converges at (alternating series) and diverges at .
  • converges everywhere (uniformly over bounded subsets). (: indeed , so .)

Power series form a ring (we can add and multiply them); facts about sums and products of numerical series imply that, where convergent, the sum/product as series is indeed equal to the sum/product as functions.

Notes not reached in this lecture, covered at the start of Lecture 20

Differentiation in one variable

(Rudin Chapter 5 McMullen §5.)

Definition 19.9. is differentiable at if i.e. such that .

Proposition 19.10. differentiable at continuous at . (The converse is false, e.g. at .)

Proof. where the first factor as and the second as ; since multiplication is continuous, . ∎

The usual rules of calculation hold: derivatives of , , …; the chain rule (see Rudin p. 104–105).

Example 19.11.

  • (), . For , . is continuous but not differentiable at ().
Graph of x sin(1/x), oscillating faster and faster near the origin between the dashed lines y = x and y = -x.
  • , : is differentiable () but is not continuous at .
Graph of x^2 sin(1/x), oscillating near the origin between the dashed parabolas y = x^2 and y = -x^2.
  • is continuous (the series converges uniformly, since converges) and nowhere differentiable! (See also Rudin 7.18 for a related example.)

Theorem 19.12 (Mean value theorem). differentiable such that .

This follows logically from easier results:

  1. If has a local max (or min) at (i.e. a max of ) and is differentiable at , then . (Because is for and for : take the limit as from the left and from the right.)
  2. If is differentiable and , then such that . This is clear if is constant; otherwise look at the max or min of over and apply (1).
  3. The mean value theorem: apply (2) to .

Corollary 19.13 (Mean value inequality). If for all , then .

Generalization:

Theorem 19.14 (Taylor’s theorem). Let be times differentiable. The degree Taylor polynomial of at is Then such that

Proof.

  • Subtracting from both sides, we can reduce to the case (and ).
  • Let . Then , and we still have .
  • Now: the mean value theorem for : such that . The mean value theorem for : such that . And so on until such that , i.e.

∎

Remark.

  • We can compare to by applying the theorem to instead!
  • As with the mean value inequality: a bound gives a bound over .

Remark. There exist nonzero functions whose Taylor polynomials are all zero! E.g. , : (all derivatives exist) and for all , so the Taylor polynomials are all zero! The Taylor series of at converges, but ! (In other examples, it can also have , i.e. never be convergent for .)