Chapter 27: The Edge of Convergence

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27.1 Alternating Series

27.2 Telescoping Series - CD

Section Summary
A series of the form collapses like a telescope.

Recall the idea of the Fundamental Theorem of Integral Calculus: In order to find

we first find an antiderivative, F[x] such that its differential satisfies dF[x]=f[x]dx. The increment equation for this new function says

so, the defining sum for the integral collapses or "telescopes,"

Sometimes, we can use this idea to find the sum of an infinite series. We cannot antidifferentiate, but we can occasionally find a difference,


The partial sum

Exercise set CD-27.1

  1. Telescoping Series
    Find the sum of the series
    • 1.
    • 2.

      Series we know can be used to estimate the error in series we do not know.

    • The Next Term Is a Bad Estimate
      How fast does the series

      converge? Use the estimates

      and sum the telescoping terms. The error series satisfies

      How much is the difference between these estimates of error? This difference gives us

      with no more than ???

      If we sum 100 terms of the series of terms , how much is the error? How does the error compare with the next term, ?

27.3 Integrals Compared to Series - CD

Section Summary
A way to estimate series above or below is to compare them with integrals.

Example CD-27.1 Comparison of the Harmonic Series and Natural Log Integral

The continuous function decreases, so it stays below for . This means that one term of the series satisfies


Figure CD-27.1 shows this estimate graphically.

Figure CD-27.1: An integral below

This shows that the harmonic series diverges.

Example CD-27.2 Integral Comparison for

An estimate of series with integrals can be used to prove convergence. We know is decreasing, so we have

This makes

In fact, the error for the series satisfies

Figure CD-27.2 shows this estimate graphically.

Figure CD-27.2: An integral above

Compare this to Exercise CD-27.2.2.

The idea of this section can be summarized as follows:

Theorem CD-27.1 Integral Comparison Suppose that f[x] is a positive continuous function, defined and decreasing for . The series of positive decreasing terms, ak=f(k),


Exercise set CD-27.2

  1. (Computer Exercise) Log Only Goes to Infinity Reluctantly Run the computer program LogGth, and see how slowly the harmonic series grows. Although it does tend to infinity, it certainly takes its time...

  2. Lower Integral Comparison
    Test the following series for divergence by squeezing an integral below them:
    • 1.
    • 2.

    • Upper Integral Comparison
      Test the following series for convergence by squashing them below an integral:
      • 1.
      • 2.

        The two previous simple integral comparison exercises generalize as follows:

      • Series with Powers
        Show that the following series diverge if and converge if p>1:
        • 1.
        • 2.
        • 3.

          • 1. The series converges. Why?
          • 2. Does converge?

27.4 Limit Comparisons - CD

Section Summary
Each time we learn a new convergent or divergent series, we can use it to compare to many other series.

Example CD-27.3

We know from Exercise CD-27.3.4 above that 1+1/2p+...+1/np+... converges for any p>1. We also know from the rate of growth of log that

The series

must therefore converge, because so that eventually

and, from that point on,

Example CD-27.4

We know from the rate of growth of log that

so, in particular, for sufficiently large n, for and

The harmonic series diverges and the tail of this series is larger,

When the limit of the ratio of the terms of two series are non-zero, they represent the same "order of infinitesimal" and thus converge or diverge together.

Theorem CD-27.2 Limit Comparison of Series Suppose the sequences ak and bk satisfy . Then
  • 1. If converges, so does .
  • 2. If diverges, so does .

We can also use each new numerical series in function estimates. For example, the Fourier series

is convergent absolutely and uniformly because

and 1+1/9+1/25+... converges.

Exercise set CD-27.3

  1. General Comparison
    Test the following series for convergence or divergence:
    • 1.
    • 2.
    • 3.
    • 4.
    • 5.
    • 6.

27.5 Fourier Series - CD

Section Summary
Fourier series arise in many mathematical and physical problems.

The series

converges to the function that equals |x| for and is then repeated periodically. Some approximating graphs are:

Figure CD-27.3: Fourier series for f[x]=|x|

The projects on series show you the simple formula for Fourier coefficients and give the interesting convergence theorem for functions. Fourier series can converge delicately. For example, the identity

is a valid convergent series for . However, the Weierstrass majorization does not yield a simple convergence estimate, because

is a useless upper estimate by a divergent series. This series converges but not uniformly, and its limit function is discontinuous because repeating x periodically produces a jump at as follows:

Figure CD-27.4: Fourier series for f[x]=x

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