tests for series convergence/divergence

An infinite series converges if there is a target value T so that for any L<T and any M>T, all the partial sums from some point on lie between L and M.

Cauchy’s approach

This avoids the unreliable rearranging of terms of possibly divergent series, and works instead with finite sums. e.g. in dealing with \dfrac{1}{1-x}=1+x+x^2+x^3\dots\quad :

    \begin{align*} 1&=1-x+x-x^2+x^2-\dots-x^n+x^n \\ &=(1-x)+x(1-x)+x^2(1-x)+\dots+x^{n-1}(1-x)+x^n \\ \fr{1-x}&=1+x+x^2+\dots+x^{n-1}+\frac{x^n}{1-x} \\ &\text{   }1+x+x^2+\dots+x^{n-1}=\fr{1-x}-\frac{x^n}{1-x} \end{align*}

The finite series differs from the target series T=\dfrac{1}{1-x} by

    \[\frac{x^n}{1-x}\]

If we take a value larger than T, is this finite sum eventually below it? If we take a value smaller than T, is this finite sum eventually above it? The value of this series is \dfrac{1}{1-x} if and only if we can make the difference as close to 0 as we wish by putting a lower bound on n. This happens precisely when |x| < 1. Cauchy’s analysis shows that the equation needs to carry a restriction:

    \[\fr{1-x}=1+x+x^2+x^3\dots \qquad \text{ when }|x|<1.\]


Test for Divergence

If \lim\limits_{n\to\infty} a_n does not exist or if \lim\limits_{n\to\infty} a_n\neq 0, then the series \sum\limits_{n=1}^\infty a_n is divergent.

The Integral Test

If f is a continuous, positive, decreasing function on [1,\infty), and a_n=f(n), then the series \sum\limits_{n=1}^\infty a_n is convergent if and only if the improper integral \int_1^\infty f(x) \;\text{d}x is convergent.

The Comparison Test

Suppose \sum a_n and \sum b_n are series with positive terms.
(i) If \sum b_n is convergent and a_n \les b_n for all n, then \sum a_n is also convergent.
(i) If \sum b_n is divergent and a_n \ges b_n for all n, then \sum a_n is also divergent.

Mostly the series \sum a_n is compared with a p-series or a geometric series.

The Limit Comparison Test
Suppose \sum a_n and \sum b_n are series with positive terms. If

    \[\lim\limits_{n\to\infty} \frac{a_n}{b_n}=c\]

where c is a finite number and c>0, then either both series converge or both diverge.

Alternating Series Test

If the alternating series

    \[\sum\limits_{n=1}^\infty (-1)^{n-1} b_n=b_1-b_2+b_3-b_4+b_5-b_6+\dots \qquad b_n>0\]

satisfies +++ (i) b_{n+1} \les b_n\qquad for all n
++++++++++ (ii) \lim\limits_{n\to\infty} b_n=0
then the series is convergent.

The Ratio Test

The Root Test

Strategy for testing series

references

David Bressoud – A Radical Approach to Real Analysis
James Stewart – Calculus: Early Transcendentals, ch.11

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