Calculus How To

Conditional Convergence: Definition, Examples

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Conditional convergence is a special kind of convergence where a series is convergent (i.e. settles on a certain number) when seen as a whole. However, there’s a catch:

  • The sum of its positive terms goes to positive infinity and
  • The sum of its negative terms goes to negative infinity.

It has a very special property, called the Riemann series theorem, that says that it can be made to converge to any desired value—or to diverge—by simple rearrangement of the terms.

One Way to Identify a Conditionally Convergent Series

In order to find out if a series is conditionally convergent:

  1. Find out if the series converges, then
  2. Determine it isn’t absolutely convergent.
    1. The Alternating Series Test tells us that if the terms of the series alternates in sign (e.g. -x, +x, -x…), and each term is bigger than the term after it, the series converges.
    2. Take the absolute values of the alternating (converging) series. If the new (all positive term) series converges, then the series is absolutely convergent. If that new series is not convergent, the original series was only conditionally convergent.

Example of Conditional Convergence

One example of a conditionally convergent series is the alternating harmonic series, which can be written as:
conditional convergence example
It converges to the limit—ln 2— conditionally, but not absolutely; make a new series by taking the absolute value of each of the terms and your new series will diverge.

Understanding the Riemann Series Theorem

It might seem counter-intuitive that a series can be made to converge to anything just by rearranging the terms. But if you have a well-defined limit you want it to approach, all you need to do is:

  1. Take enough positive terms to just barely exceed the desired limit, then
  2. Add enough negative terms to go below the desired limit, then
  3. Continue in this way.

Since all terms of the original series go to zero, the new, rearranged series will converge to the limit you chose.

As an example of the Riemann series consider the alternating harmonic series, which we looked at above. As written, it converges to ln2. But can we make it converge to half of that, (ln2)/2. The ordinary way, it would be written

1-1/2 + 1/3 – 1/4 +….

Every other term is negative. But if we arrange it as (one positive term) + (two negative terms), we get this:

1 – 1/2 -1/4 + 1/3…

We can rewrite this as:

Which is one half of what the original series converged to.


Absolute and Conditional Convergence. Retrieved from on December 22, 2018.

Stephanie Glen. "Conditional Convergence: Definition, Examples" From Calculus for the rest of us!

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