Weird stuff on infinite numerical sequences in a Soviet book

In summary, the author defines an infinite numerical sequence as a rule assigning every natural number to a definite term in the sequence, but 1 does not correspond with anything, at least for the given expression. But he sees what you mean.
  • #1
inthenickoftime
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TL;DR Summary
It looks like an undefined operation
The book is Calculus: Basic Concepts for High School
on the first page you are given the following sequence:

1, -1, 1/3, -1/3, 1/5, -1/5, 1/7, -1/7, ...

several pages later the rule is given:

Untitled.jpg

in the second rule, for the first term in the sequence, the coefficient of one of the terms is 1/0. How legitimate is this?
 
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  • #2
Start with ##n=2## instead of ##n=1## or make a shift. For most purposes in analysis, leaving out finitely many terms of a sequence doesn't matter since we are mostly interested in the tail of the sequence.
 
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  • #3
Math_QED said:
Start with ##n=2## instead of ##n=0## or make a shift. For most purposes in analysis, leaving out finitely many terms of a sequence doesn't matter since we are mostly interested in the tail of the sequence.
It still feels weird. On the previous page the author gives his definition of an infinite numerical sequence as a rule assigning every natural number to a definite term in the sequence, but 1 doesn't correspond with anything, at least for the given expression. But I see what you mean
 
  • #4
inthenickoftime said:
It still feels weird. On the previous page the author gives his definition of an infinite numerical sequence as a rule assigning every natural number to a definite term in the sequence, but 1 doesn't correspond with anything, at least for the given expression. But I see what you mean
You can define a sequence on any infinite subset of the natural numbers. In other words, you can do things like:

Define the sequence for ##n = n_0, n_0 +1, n_0 + 2 \dots##, where ##n_0## can be any starting number.

Define the sequence for odd, even or square or prime ##n##. For example, you could define a sequence as ##a_n##, where ##n## is prime.

That said, in all these cases you ought to make clear the restriction on ##n##. E.g. ##n \ge 2## or ##n## even or ##n## prime etc.
 
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  • #5
PeroK said:
You can define a sequence on any infinite subset of the natural numbers. In other words, you can do things like:

Define the sequence for ##n = n_0, n_0 +1, n_0 + 2 \dots##, where ##n_0## can be any starting number.

Define the sequence for odd, even or square or prime ##n##. For example, you could define a sequence as ##a_n##, where ##n## is prime.

That said, in all these cases you ought to make clear the restriction on ##n##. E.g. ##n \ge 2## or ##n## even or ##n## prime etc.
No restrictions were given. You can check the book online for free, it was rewritten in modern style. Too bad as I was really looking forward to it.
 
  • #6
The yn pattern listed there only applies to n>1, for y1 it does not apply. If you care about the absolute value of the limit you need to consider y1 separately, if you only care about convergence you do not.
 
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Related to Weird stuff on infinite numerical sequences in a Soviet book

1. What is the significance of the "weird stuff" in the Soviet book?

The "weird stuff" in the Soviet book refers to the discovery of infinite numerical sequences that exhibit strange or unexpected patterns. These sequences challenge traditional mathematical concepts and have sparked interest among mathematicians and scientists.

2. How were these infinite numerical sequences discovered?

These sequences were discovered through extensive mathematical analysis and experimentation. The Soviet book, "Weird Stuff on Infinite Numerical Sequences," details the methods used to uncover these patterns and provides examples of the sequences found.

3. What implications do these infinite numerical sequences have?

The discovery of these infinite numerical sequences has implications for our understanding of mathematics and the universe. They challenge traditional theories and open up new possibilities for further research and exploration.

4. Are these infinite numerical sequences relevant to real-world applications?

While these sequences may seem abstract and theoretical, they have the potential to impact real-world applications. They could potentially be used in fields such as cryptography, data compression, and computer science.

5. What does the study of these infinite numerical sequences tell us about the Soviet scientific community?

The study of these infinite numerical sequences in a Soviet book reflects the advanced level of mathematical research and innovation within the Soviet scientific community. It also highlights the importance of collaboration and sharing knowledge among scientists in different countries.

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