Convergent and Divergent Series

Excerpts

• Convergent and Divergent Series

  On the other hand, a series is said to be *divergent* when the sum of a number of terms may be made to surpass any quantity, however great.

• Convergent and Divergent Series

  We have introduced the new terms, $\dfrac{b}{a}$, $\dfrac{c}{b}$, etc., or the ratios which the several terms of the original series bear to those immediately preceding.

• Convergent and Divergent Series

  A series is said to be *convergent* when the sum of its terms tends towards some limit; that is, when, by taking any number of terms, however great, we shall never exceed some certain quantity. On the other hand, a series is said to be *divergent* when the sum of a number of terms may be made to surpass any quantity, however great.

• Convergent and Divergent Series

  A series cannot be convergent, unless its separate terms decrease, so as, at last, to become less than any given quantity.

• Convergent and Divergent Series

  And the terms of a series may at first increase and afterwards decrease, being apparently divergent for a finite number of terms, and convergent afterwards. It will only be necessary to consider the latter part of the series.

• Convergent and Divergent Series

  It may be shown (1) that if the terms of the series $\dfrac{b}{a}$, $\dfrac{c}{b}$, $\dfrac{d}{c}$, etc., come at last to be less than unity, and afterwards either continue to approximate to a limit which is less than unity, or decrease without limit, the series $a + b + c + \etc.$, is convergent; (2) if the limit of the terms $\dfrac{b}{a}$, $\dfrac{c}{b}$, etc., is either greater than unity, or if they increase without limit, the series is divergent.

• Convergent and Divergent Series

  But since $\dfrac{l}{k}$ is less than unity, the first can never surpass $k × \dfrac{1}{1 - \dfrac{l}{k}}$, or $\dfrac{k^{2}}{k - l}$, and is convergent; the second is therefore convergent.

• Convergent and Divergent Series

  (2) The second theorem on the divergence of series we leave to the student’s consideration, as it is not immediately connected with our object.

Equations

• Convergent and Divergent Series

  a\left(1 + \frac{b}{a} + \frac{c}{b}\, \frac{b}{a} + \frac{d}{c}\, \frac{c}{b}\, \frac{b}{a} + \etc.\right)

  The series a + b + c + d + ... can be rewritten as a times a series whose terms are successive ratios of the original terms.

• Convergent and Divergent Series

  k + l + m + \etc. = k\left(1 + \frac{l}{k} + \frac{m}{l}\, \frac{l}{k} + \etc.\right)

  The same rewriting holds when the series is taken from any term k onward: the tail sum equals k times a series of successive ratios.

• Convergent and Divergent Series

  \dfrac{l}{k} > \dfrac{m}{l} > \dfrac{n}{m}

  The successive ratios of the terms are assumed to decrease, so each ratio is greater than the next.

Problems

No exercises in this chapter.