Rearranging terms of a conditionally convergent series

In summary, the concept of rearranging terms in a series going towards infinity can cause a change in the sum. This is due to the idea of shifting elements in a specific direction to make the sum diverge. Wikipedia provides examples and explanations of this concept.
  • #1
Nikitin
735
27
Why the heck will the sum of such a series going towards infinity change if its terms are re-arranged? My book omits the proof, and without it this claim makes no sense to me.

Can somebody provide an example of such a series, and maybe some light explanation (I'm way too exhausted for heavy stuff)?
 
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  • #2
Wikipedia has examples
The basic idea is to shift many elements with one specific sign away ("to the right") to get the sum to diverge in the chosen direction.
 
  • #3
The idea is something like this: pick your favorite number, L. Add up some positive terms until you get something that is bigger than L. Then add some negative terms until you get something that is less than L, then add some positives until you get bigger than L, and so on. Since the series is not absolutley congergent, you will always be able to do this.
 

FAQ: Rearranging terms of a conditionally convergent series

What is a conditionally convergent series?

A conditionally convergent series is a mathematical series in which the sum of the terms converges, but the series does not converge absolutely. This means that the sum of the terms will approach a finite value, but the order in which the terms are added can greatly affect the final sum.

What does it mean to rearrange the terms of a conditionally convergent series?

When we rearrange the terms of a conditionally convergent series, we are changing the order in which the terms are added. This can result in a different sum for the series, as the terms may now be added in a different order.

Why is rearranging the terms of a conditionally convergent series important?

Rearranging the terms of a conditionally convergent series is important because it can demonstrate the concept of conditional convergence. It also shows that the sum of a series can be greatly affected by the order in which the terms are added, and highlights the difference between absolute and conditional convergence.

Can rearranging the terms of a conditionally convergent series change the final sum?

Yes, rearranging the terms of a conditionally convergent series can change the final sum. This is because the terms may be added in a different order, causing the series to converge to a different value. This demonstrates the concept of conditional convergence, where the sum of a series is dependent on the order of the terms.

Are there any rules or guidelines for rearranging the terms of a conditionally convergent series?

Yes, there are some rules and guidelines for rearranging the terms of a conditionally convergent series. One important rule is the Riemann rearrangement theorem, which states that a conditionally convergent series can be rearranged to converge to any desired value, or even diverge. It also states that the sum of the series will be different for different rearrangements. Additionally, there are techniques such as grouping and partial sums that can be used to rearrange a series without changing its sum.

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