Absolute Convergence
If a series has some positive and some negative terms, there are a couple of things that one might ask. The first is
1) does the series converge? Another question, the motivation for which is less obvious, is
2) does the series converge if we take the absolute values of its terms?
If the first answer is yes, the second can be yes or no. It turns out that if this second answer is yes, the series behaves much like a finite sum, i.e. it behaves well.
Definition: Let $\sum a_n$ be a series.
If the series $ {\sum \left|a_n\right|}$ converges, then we say that $\sum a_n$ is
absolutely convergent.
If $\sum a_n$ converges but $\sum \left|a_n\right|$ doesn't, then we say that $\sum a_n$ is conditionally convergent.
Example: Consider the alternating harmonic series $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} = 1 - \frac12 + \frac13 - \frac14+\cdots.$$It
converges (we saw this previously by using the AST). The series with the absolute values of its terms, which is the harmonic series $\sum \frac{1}{n}$,
diverges ($p$-series with $p\le 1$).
Since the series converges, but not in absolute value, we say it is conditionally convergent.
One fact, said in two ways
If $\sum \left|a_n\right|$ converges, then $\sum a_n$ converges.
(Absolutely convergent $\Longrightarrow$ convergent.) If $\sum a_n$ does not converge, then $\sum\left|a_n\right|$ will not converge.
(Divergent $\Longrightarrow$ not absolutely convergent.)
Example: Consider the alternating $p$-series, with $p=2$, $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^2} = 1 - \frac14 + \frac19 - \frac1{16}+\cdots.$$Since the series with the absolute values of the terms of our series, $\sum\frac{1}{n^2}$, is a convergent $p$-series, our
series is absolutely convergent. By the fact above, this means
it is also convergent. It is not conditionally convergent.
Be careful with these terms
Conditional convergence of a series means it is
convergent but not absolutely convergent. If we are told that a series is
convergent, we do not know
a priori whether it is
conditionally convergent or absolutely convergent. It is
one or the other, but not both.
Every series is
either divergent, conditionally convergent, or absolutely convergent, but it is
only one of these things.
Justification of the fact above, and some examples, are discussed in the video.
FAQs
In mathematics, an alternating series is an infinite series of the form. or. with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
What is meant by alternating series? ›
In mathematics, an alternating series is an infinite series of the form. or. with an > 0 for all n. The signs of the general terms alternate between positive and negative. Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
How do you know if a series is alternating? ›
8.4. 5 Summary
- An alternating series is a series whose terms alternate in sign. ...
- The sequence of partial sums of a convergent alternating series oscillates around the sum of the series if the sequence of th terms converges to 0.
Those three conditions are: + bn Condition (1): The series is an alternating series (it has the form where by > 0); Condition (2): The absolute value of each term is less than or equal to that of the preceding term (the statement br+1 <bn is true for all n); and Condition (3): The terms shrink toward zero for large n, ...
What is a series that alternates signs? ›
An alternating series is a series where an flips sign every turn. For instance, ∞∑n=1(−1)nn=1−12+13−14+….
What is an example of an alternating sequence? ›
Examples:
- an=(−12)n. This sequence would have terms: −12;14;−18;116;...
- bn=(−1)n . This sequence would have terms: −1;1;−1;1;...
- cn=(−1)n⋅n. This sequence would have terms: −1;2;−3;4;...
Definitions of alternating. adjective. occurring by turns; first one and then the other.
What happens if an Alternating Series Test fails? ›
What do you do if the Alternating Series Test fails? In most cases, an alternation series ∞∑n=0(−1)nbn fails Alternating Series Test by violating limn→∞bn=0 . If that is the case, you may conclude that the series diverges by Divergence (Nth Term) Test.
When can you not use the Alternating Series Test? ›
We cannot remove a finite number of terms to make decreasing, therefore we cannot apply the alternating series test. Keep in mind that this does not mean we conclude the series diverges; in fact, it does converge. We are just unable to conclude this based on the alternating series test.
What is the general formula for the alternating series? ›
For an alternating series ∞∑n=1(−1)n+1bn, if bk+1≤bk for all k and bk→0 as k→∞, the alternating series converges.
An alternating series cannot diverge to ∞ or to −∞ since it's consecutive terms are of opposite sign. This series converges/ oscillates if and only if it's sequence of partial sums converges/ oscillates . For example , the alternating series Σ∞n=1(−1)n+11n Σ n = 1 ∞ ( − 1 ) n + 1 1 n converges to loge(2).
What is a repeating series called? ›
In mathematics, a periodic sequence (sometimes called a cycle or orbit) is a sequence for which the same terms are repeated over and over: a1, a2, ..., ap, a1, a2, ..., ap, a1, a2, ..., ap, ...
What are the two types of series? ›
The arithmetic series is represented by a + (a+d) + (a+2d) + (a+3d) + … The geometric series is represented by a + ar + ar2 + ar3 + ….+ arn-1+ ..
What is alternating current in simple words? ›
Similar term(s): AC & DC. Definition: Alternating Current (AC) is a type of electrical current, in which the direction of the flow of electrons switches back and forth at regular intervals or cycles. Current flowing in power lines and normal household electricity that comes from a wall outlet is alternating current.
What is meant by alternating system? ›
Alternating current (AC) is an electric current that periodically reverses direction and changes its magnitude continuously with time, in contrast to direct current (DC), which flows only in one direction.
What is the general term of alternating series? ›
To make a series alternate, you generally stick in a factor of (−1)n or (−1)n+1. In your case, the general term could be (−1)n+1⋅5n+8 for n=1,2,3…
Does alternating mean changing? ›
verb (used without object),al·ter·nat·ed, al·ter·nat·ing. to interchange repeatedly and regularly with one another in time or place; rotate (usually followed by with): Day alternates with night. to change back and forth between conditions, states, actions, etc.: He alternates between hope and despair.
