5.3.1 Convergence Because Of Cancellation
So far, the tests for convergence have been applied to non negative terms only. Sometimes, a
series converges, not because the terms of the series get small fast enough, but because of
cancellation taking place between positive and negative terms. A discussion of this involves
some simple algebra.
be sequences and let
Then if p < q
This formula is called the partial summation formula.
Theorem 5.3.1 (Dirichlet’s test) Suppose An ≡∑
k=1nak is bounded and
limn→∞bn = 0, with bn ≥ bn+1 for all n. Then
Proof:This follows quickly from Theorem 5.1.7. Indeed, letting
the partial summation formula above along with the assumption that the bn
and by assumption, this last expression is small whenever p and q are sufficiently large.
Definition 5.3.2 If bn > 0 for all n, a series of the form ∑
k−1bk is known as an alternating series.
The following corollary is known as the alternating series test.
Corollary 5.3.3 (alternating series test) If limn→∞bn = 0, with bn ≥ bn+1, then
Proof:Let an =
Then the partial sums of ∑
are bounded and so Theorem
In the situation of Corollary 5.3.3 there is a convenient error estimate available.
Theorem 5.3.4 Let bn > 0 for all n such that bn ≥ bn+1 for all n and
limn→∞bn = 0 and consider either ∑
nbn or ∑
See Problem 8 on Page 244 for an outline of the proof of this theorem along with another
way to prove the alternating series test.
Example 5.3.5 How many terms must I take in the sum, ∑
From Theorem 5.3.4, I need to find n such that
1 is the desired
value. Thus n
= 3 and so
Definition 5.3.6 A series ∑
an is said to converge absolutely if ∑
converges. It is said to converge conditionally if ∑
fails to converge but ∑
Thus the alternating series or more general Dirichlet test can determine convergence of
series which converge conditionally.