The Unapologetic Mathematician

Mathematics for the interested outsider

Convergence Tests for Infinite Series

Now that we’ve seen infinite series as improper integrals, we can immediately import our convergence tests and apply them in this special case.

Take two sequences a_k and b_k with b_k\geq a_k\geq0 for all k beyond some point N. Now if the series \sum_{k=0}^\infty a_k diverges then the series \sum_{k=0}^\infty b_k does too, and if the series \sum_{k=0}^\infty b_k converges to b then the series of \sum_{k=0}^\infty a_k converges to a\leq b.

[UPDATE]: I overstated things a bit here. If the series of b_k converge, then so does that of a_k, but the inequality only holds for the tail beyond N. That is:

\displaystyle\sum\limits_{k=N}^\infty a_k\leq\sum\limits_{k=N}^\infty b_k

but the terms of the sequence a_k before N may, of course, be so large as to swamp the series of b_k.

If we have two nonnegative sequences a_k and b_k so that \lim\limits_{k\rightarrow\infty}\frac{a_k}{b_k}=c\neq0 then the series \sum_{k=0}^\infty a_k and \sum_{k=0}^\infty b_k either both converge or both diverge.

We read in Cauchy’s condition as follows: the series \sum_{k=0}^\infty a_k converges if and only if for every \epsilon>0 there is an N so that for all n\geq m\geq N the sum \left|\sum_{k=m}^n a_k\right|<\epsilon.

We also can import the notion of absolute convergence. We say that a series \sum_{k=0}^\infty a_k is absolutely convergent if the series \sum_{k=0}^\infty|a_k| is convergent (which implies that the original series converges). We say that a series is conditionally convergent if it converges, but the series of its absolute values diverges.

About these ads

April 25, 2008 - Posted by | Analysis, Calculus

6 Comments »

  1. […] on and the fact that is decreasing imply that , and the series clearly converges. Thus by the comparison test, the series converges absolutely, and our result follows. This is called Dirichlet’s test […]

    Pingback by Dirichlet’s and Abel’s Tests « The Unapologetic Mathematician | May 1, 2008 | Reply

  2. […] if we set , this tells us that . Then the comparison test with the geometric series tells us that […]

    Pingback by The Ratio and Root Tests « The Unapologetic Mathematician | May 5, 2008 | Reply

  3. […] don’t get smaller. And we know that must be zero, or else we’ll have trouble with Cauchy’s condition. With the parentheses in place the terms go to zero, but when we remove them this condition can […]

    Pingback by Associativity in Series II « The Unapologetic Mathematician | May 7, 2008 | Reply

  4. […] if is only conditionally convergent, I say that we can rearrange the series to give any value we want! In fact, given (where these […]

    Pingback by Commutativity in Series I « The Unapologetic Mathematician | May 8, 2008 | Reply

  5. […] of Complex Series Today, I want to note that all of our work on convergence of infinite series carries over — with slight modifications — to complex […]

    Pingback by Convergence of Complex Series « The Unapologetic Mathematician | August 28, 2008 | Reply

  6. […] the series of the converges, Cauchy’s condition for series of numbers tells us that for every there is some so that when and are bigger than , . But now when we […]

    Pingback by Uniform Convergence of Series « The Unapologetic Mathematician | September 9, 2008 | Reply


Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Follow

Get every new post delivered to your Inbox.

Join 393 other followers

%d bloggers like this: