Convergent Series: Am I and The Book Wrong or is Wolfram?

In summary, the conversation discusses the convergence of the series (1+x)^n for x<1, as stated in a book. However, when plugging in n=-1, WolframAlpha states that the series does not converge. The conversation then explores using x=1/y, where y>1, to expand the series and show that it is convergent. It also mentions the importance of absolute convergence in the process. It concludes by questioning whether the book or WolframAlpha is incorrect, with the assumption that the book and WolframAlpha are correct in their statements.
  • #1
Miike012
1,009
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In my book it says that the series (1+x)^n converges for x<1.

However I put n = -1 and wolfram says that the series does not converge.

However if I let x = 1/y where y>1

then the expansion of (1+1/y)^-1 is equal to: (which I will define as (SERIES 1))
1 - y + (1/y)2 - (1/y)3 + (1/y)4 - ...

= 1 + ( (1/y)3 + (1/y)5 + ... ) - ( (1/y)2 + (1/y)4 + ...)

The series (1/y)3 + (1/y)5 + ... (1/y)3 + 2n + .. is equal to (which I will define as (SERIES 2))
Ʃ(1/y)3 + 2n where n→∞ and 1≤n<∞.
I also know that the sum of the (SERIES 2) is less than the series 1 + 1/4 + 1/8 + 1/16 + ... which is convergent therefore (SERIES 2) is convergent.

Also from (SERIES 1) I know that the sum is positive and therefore the series
( (1/y)2 + (1/y)4 + ...) is less than (SERIES 2) + 1 and therefore as the number of terms approaches ∞ the series
( (1/y)2 + (1/y)4 + ...) which is positive is less than a [(finite number) + 1] which is a finite number and therefore is convergent.

∴Therefore the (SERIES 1) is convergent.


Am I and the book wrong or is wolfram?
 

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  • #2
Miike012 said:
In my book it says that the series (1+x)^n converges for x<1.
(1+x)^n is not a series.

If you want to sum that over natural n, it is pointless to set n to anything special (in particular, negative values).

Do you want to get the taylor expansion at x=0? That converges for x<1 for all n, and for all x for integer n (including 0 if we define 00=1).


1 - y + (1/y)2 - (1/y)3 + (1/y)4 - ...

= 1 + ( (1/y)3 + (1/y)5 + ... ) - ( (1/y)2 + (1/y)4 + ...)
That step requires absolute convergence, so
1 + |1/y| + |(1/y)2| + |(1/y)3| ...
has to converge, too.

I also know that the sum of the (SERIES 2) is less than the series 1 + 1/4 + 1/8 + 1/16 + ... which is convergent therefore (SERIES 2) is convergent.
That is true for some y only.

Am I and the book wrong or is wolfram?
I would expect that the book and WolframAlpha are right. What did you use as query for WA?
 

FAQ: Convergent Series: Am I and The Book Wrong or is Wolfram?

What is a convergent series?

A convergent series is a mathematical concept that refers to a sequence of numbers that approaches a specific limit as the number of terms increases. In other words, the sum of the terms in a convergent series will eventually reach a finite value.

How do you determine if a series is convergent or divergent?

There are several methods for determining if a series is convergent or divergent, including the integral test, comparison test, and ratio test. These tests involve evaluating the behavior of the terms in the series and comparing them to known convergent or divergent series.

What is the significance of Wolfram's role in "Convergent Series: Am I and The Book Wrong or is Wolfram?"

Wolfram is a reference to Stephen Wolfram, a prominent mathematician and physicist who developed the computational software Wolfram Alpha. In the context of this question, Wolfram represents a source of information and authority on the topic of convergent series.

What are some real-world applications of convergent series?

Convergent series have many practical applications, including in finance, engineering, and physics. For example, in finance, convergent series are used to calculate compound interest over time. In engineering, convergent series are used in the analysis of circuits and signal processing. In physics, convergent series are used to model and predict the behavior of physical systems.

What should I do if I believe there is an error in "Convergent Series: Am I and The Book Wrong or is Wolfram?"

If you believe there is an error in "Convergent Series: Am I and The Book Wrong or is Wolfram?", you can try reaching out to the author or publisher of the book to report the potential error. You can also consult other sources or experts in the field to verify the information and potentially provide a correction. It is important to critically evaluate information and seek multiple sources to ensure accuracy.

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