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 .

**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 :

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

satisfies ~~+++~~ (i) for all

~~++++++++++~~ (ii)

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