Exploring Patterns in Mathematical Series and Sequences

In summary, the conversation is about finding three mathematical series for specific patterns, including one that involves a subset of natural numbers and the "Hilbert's Hotel" paradox. The suggested solutions for the first two patterns involve using the properties of odd and even numbers and periodicity, while the third pattern relates to the philosophical concept of infinity.
  • #1
real analyst
10
0

Homework Statement



I need to find three mathematical series, for the following patterns

1) Sn= 1/n if n is odd, or 1 of n is even.

2) Sn= 0,1,0,.5,1,0,1/3,2/3,0,1/4...

3) Find a sequence (Sk) which is a subset of the natural numbers in which every positive integer appears infinitely many times...that is, for each n in the Natural numbers, Sk=n for infinately many k's. any ideas?

Homework Equations



I guess they're arent any relevant equations to these. I guess yuo just have to get the pattern

The Attempt at a Solution



well, I know the first two are similar and that's about all I know. they're not p-series, geometric, alternating, or anything like that so I'm kind of stuck

The third one, well, I don't even kow what it means honestly, so can you at least give me a hint?? thanks.
 
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  • #2
For 1), try thinking along the lines of

[tex]
(-1)^n =
\begin{cases}
-1 & n\mbox{ is odd}\\
1 & n\mbox{ is even}
\end{cases}
[/tex]

For 2), try thinking along the lines of the periodicity of [tex]\sin(x)[/tex].

For 3), I'll have to refer you to the Wikipedia article on "[URL Paradox[/URL].
 
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FAQ: Exploring Patterns in Mathematical Series and Sequences

What is the difference between a series and a sequence?

A sequence is a list of numbers that follow a specific pattern or rule, while a series is the sum of a sequence. In other words, a series is the result of adding all the terms in a sequence together.

How do you identify the type of series or sequence?

To identify the type of series or sequence, you need to look at the pattern of numbers and determine if there is a common difference (arithmetic), a common ratio (geometric), or another pattern such as alternating signs or squares.

What is the formula for finding the nth term in a series or sequence?

The formula for finding the nth term in a series or sequence depends on the type of series or sequence. For arithmetic sequences, the formula is an = a1 + (n-1)d, where a1 is the first term, n is the term number, and d is the common difference. For geometric sequences, the formula is an = a1rn-1, where a1 is the first term, n is the term number, and r is the common ratio.

What is the difference between a convergent and a divergent series?

A convergent series is a series where the sum of the terms approaches a finite number as the number of terms increases. In other words, the series has a definite sum. A divergent series is a series where the sum of the terms does not approach a finite number, and therefore, the series does not have a definite sum.

How do you determine the convergence or divergence of a series?

To determine the convergence or divergence of a series, you can use various tests such as the divergence test, comparison test, ratio test, or root test. These tests involve analyzing the behavior of the terms in the series and can help determine if the series converges or diverges.

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