An infinite series converges if there is a target value so that for any and any , all the partial sums from some point on lie between and .
This avoids the unreliable rearranging of terms of possibly divergent series, and works instead with finite sums. e.g. in dealing with :
The finite series differs from the target series by
If we take a value larger than , is this finite sum eventually below it? If we take a value smaller than , is this finite sum eventually above it? The value of this series is if and only if we can make the difference as close to as we wish by putting a lower bound on . This happens precisely when . Cauchy’s analysis shows that the equation needs to carry a restriction:
Test for Divergence
If does not exist or if , then the series is divergent.
The Integral Test
If is a continuous, positive, decreasing function on , and , then the series is convergent if and only if the improper integral is convergent.
The Comparison Test
Suppose and are series with positive terms.
(i) If is convergent and for all , then is also convergent.
(i) If is divergent and for all , then is also divergent.
Mostly the series is compared with a -series or a geometric series.
The Limit Comparison Test
Suppose and are series with positive terms. If
where is a finite number and , then either both series converge or both diverge.
Alternating Series Test
If the alternating series
+++ (i) for all
then the series is convergent.
The Ratio Test
The Root Test
Strategy for testing series
David Bressoud – A Radical Approach to Real Analysis
James Stewart – Calculus: Early Transcendentals, ch.11