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Math Analysis 1

Numerical series: convergence, tests and alternating series

Partial sums, geometric, harmonic and telescoping series, the necessary condition, comparison, ratio, root and integral tests, Leibniz and absolute convergence.

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Adding infinitely many numbers sounds meaningless, and yet sometimes the result is a finite, precise number: \frac12 + \frac14 + \frac18 + \dots = 1. Numerical series study when this happens. They are the basis of Taylor and Fourier series, and therefore of a large part of signal theory.

Definition

Given a sequence a_n, the series \sum_{n=1}^{\infty} a_n is the limit of the partial sums S_N = a_1 + a_2 + \dots + a_N. If the limit is finite the series converges, if it is infinite it diverges, if it does not exist it oscillates.

Reference series

  • Geometric \sum r^n: converges if and only if |r| < 1, to \frac{1}{1-r} when starting from n = 0 (proof in the in-depth article).
  • Generalised harmonic \sum \frac{1}{n^\alpha}: converges if and only if \alpha > 1. The harmonic series \sum\frac1n diverges, though very slowly: it takes more than 10^{43} terms to exceed 100.
  • Telescoping: \sum\frac{1}{n(n+1)} = \sum\left(\frac1n - \frac{1}{n+1}\right). The partial sums equal 1 - \frac{1}{N+1}, so the series converges to 1.

A necessary condition

If \sum a_n converges, then a_n \to 0. It is only necessary: the harmonic series has terms tending to 0 and still diverges. In practice it is used the other way round: if a_n does not tend to 0, the series does not converge. For example \sum\frac{n}{n+1} diverges straight away.

Tests for series with positive terms

TestStatementWhen to use it
Comparison0 \le a_n \le b_n: if \sum b_n converges so does \sum a_nsimple bounds
Limit comparisonif a_n \sim b_n, same behaviourratios of polynomials, roots
Ratio\frac{a_{n+1}}{a_n} \to \ell: converges if \ell < 1, diverges if \ell > 1factorials, powers a^n
Root\sqrt[n]{a_n} \to \ell: converges if \ell < 1, diverges if \ell > 1terms like (\dots)^n
Integrala_n = f(n), f positive and decreasing: same behaviour as \int_1^\infty flogarithms, borderline cases

If the ratio or root test gives \ell = 1 it is inconclusive: this happens with every series of the form \sum\frac{1}{n^\alpha}, and there you need comparison.

Examples.

  • \sum\frac{n+1}{n^3 + 2}: \sim\frac{1}{n^2}, converges.
  • \sum\frac{2^n}{n!}: ratio \frac{2}{n+1} \to 0, converges (to e^2 - 1).
  • \sum\left(\frac{n}{2n+1}\right)^n: root \frac{n}{2n+1} \to \frac12, converges.
  • \sum\sin\frac1n: \sim\frac1n, diverges.
  • \sum\frac{1}{n\ln n} (from n = 2): by the integral test, \int\frac{dx}{x\ln x} = \ln(\ln x) \to \infty, diverges.

Alternating series and absolute convergence

Leibniz test: if b_n is positive, decreasing and tends to 0, the series \sum (-1)^n b_n converges, and the error made by stopping after N terms is less than the first omitted term. So 1 - \frac12 + \frac13 - \frac14 + \dots converges (to \ln 2), even though the series of absolute values, the harmonic series, diverges.

A series is absolutely convergent if \sum |a_n| converges, and then \sum a_n converges too. The difference matters: absolutely convergent series can be rearranged without changing the sum, while merely convergent ones cannot.

Towards series of functions

Series become really useful when the terms depend on x. Taking the Taylor expansion to infinity gives, for instance, e^x = \sum\frac{x^n}{n!} for every x. The Fourier series writes a periodic signal as a sum of sinusoids: the square wave of amplitude 1 is \frac4\pi\left(\sin\omega t + \frac13\sin3\omega t + \frac15\sin5\omega t + \dots\right). The coefficients \frac1n decrease slowly, which is why a square wave contains harmonics up to very high frequencies and needs a lot of bandwidth to be transmitted without distortion.

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