Counterintuitive Convergent Series

In summary, the conversation discusses a strange claim made in a textbook about a convergent series in mathematical analysis. The claim states that 1 + 2 + 4 + 8 + ... equals -1, which the speaker finds confusing. They consider two possible explanations - either the claim is false or it only applies in the realm of mathematics. They also mention the importance of considering the allowed values for q in the expression.
  • #1
Moridin
692
3
[SOLVED] Counterintuitive Convergent Series

Homework Statement



One of my new textbooks in mathematical analysis makes a very strange claim (not sure if it was a true claim or some random historical anecdote) for a convergent series in one of its short sections on the history of mathematics, which I am baffled about.

Homework Equations



[tex]1 + q + q^2 + q^3 + ... = \frac{1}{1-q}[/tex]

[tex]q = 2 \rightarrow [/tex]

[tex]1 + 2 + 4 + 8 +... = -1[/tex]

?

The Attempt at a Solution



It can either be one of two explanations in my mind; it is either completely false in some way (undefined or misapplication of a theorem or wrong approach) or only applies in the realm of mathematics and you do not get -1 apples if you keep adding them together.

Thank you for your time.
 
Last edited:
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  • #2
I do hope the textbook mentions that not all values of q are permitted for that expression.
The left side converges only when |q|<1.
 
  • #3
Indeed, I finally managed to find the passage it was referring to. Thanks for your help.
 

1. What is a counterintuitive convergent series?

A counterintuitive convergent series is a mathematical series in which the sum of the terms appears to be getting larger and larger, but in reality, the series converges to a finite value. This means that even though the individual terms may be increasing, the overall sum eventually reaches a stable value.

2. How is a counterintuitive convergent series different from a regular convergent series?

In a regular convergent series, the sum of the terms decreases and approaches a finite value as the number of terms increases. However, in a counterintuitive convergent series, the sum of the terms initially increases, giving the impression that the series is divergent, but ultimately converges to a finite value.

3. What are some examples of counterintuitive convergent series?

One example is the Grandi's series, which is written as 1-1+1-1+... This series appears to be divergent, as the sum of the terms alternates between 0 and 1. However, it actually converges to the value of 1/2. Another example is the alternating harmonic series, which is written as 1-1/2+1/3-1/4+... This series also appears to be divergent, but it converges to the value of ln(2).

4. How can a counterintuitive convergent series be proven to converge?

To prove that a counterintuitive convergent series converges, one can use techniques such as the alternating series test or the Cauchy condensation test. These tests help determine if the series is alternating or if the terms decrease quickly enough for the series to converge.

5. Why is understanding counterintuitive convergent series important in science?

Counterintuitive convergent series play an important role in various scientific fields, such as physics and engineering. They can be used to model real-life phenomena, such as oscillations and electrical circuits. Understanding these series can also help scientists make accurate predictions and develop more efficient systems and processes.

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