Counterintuitive Convergent Series

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SUMMARY

The discussion centers on the mathematical analysis of the convergent series represented by the formula 1 + q + q^2 + q^3 + ... = 1/(1-q). Specifically, when q equals 2, the series yields an unexpected result of -1, which raises questions about the validity of the claim. The participants clarify that the series converges only when the absolute value of q is less than 1 (|q|<1), confirming that the textbook's assertion is misleading without proper context. This highlights the importance of understanding convergence criteria in series analysis.

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[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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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.
 
Indeed, I finally managed to find the passage it was referring to. Thanks for your help.
 

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