P-series .... Sohrab, Proposition 2.3.12 .... ....

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In summary: It seems plausible that if q \lt 0 then n^q \lt 1 ... but how do we (formally and rigorously) prove it ... especially worrying, intuitively speaking, are those values -1 \lt q \lt 0 ... ... and then how do we demonstrate(formally and rigorously) that n^q \lt 1 implies 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N} ..Theorem 1.2 in Sohrab's book states that "if q < 0, then n^q < 1".Theorem 1.2 in Sohrab's book states that "if q < 0,
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I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).

I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...

I need help with an aspect of the proof of Proposition 2.3.12 ...

Proposition 2.3.12 and its proof read as follows:

View attachment 9051
In the above proof by Sohrab we read the following:

" ... ... Now if \(\displaystyle p \leq 1\), then \(\displaystyle 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N}\) ... "My question is ... how do we know this is true ... ?

Can someone please demonstrate how to prove that if \(\displaystyle p \leq 1\), then \(\displaystyle 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N}\) ...
Help will be much appreciated ...

Peter
 

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Peter said:
" ... ... Now if \(\displaystyle p \leq 1\), then \(\displaystyle 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N}\) ... "My question is ... how do we know this is true ... ?

Can someone please demonstrate how to prove that if \(\displaystyle p \leq 1\), then \(\displaystyle 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N}\) ...

If $p\le1$ then $q=p-1<0$ and so $n^q\le1$.
 
  • #3
Olinguito said:
If $p\le1$ then $q=p-1<0$ and so $n^q\le1$.
Thanks for the reply Olinguito ...... but ... I do not follow ... can you give some more details please ...Peter
 
  • #4
If $q$ is negative, then $n^q=e^{q\ln n}\le1$ for any positive integer $n$, is that not so?
 
  • #5
Olinguito said:
If $q$ is negative, then $n^q=e^{q\ln n}\le1$ for any positive integer $n$, is that not so?
Hi Olinguito ...

It seems plausible that if \(\displaystyle q \lt 0\) then \(\displaystyle n^q \lt 1\) ... but how do we (formally and rigorously) prove it ... especially worrying, intuitively speaking, are those values \(\displaystyle -1 \lt q \lt 0\) ...

... and then how do we demonstrate(formally and rigorously) that \(\displaystyle n^q \lt 1\) implies \(\displaystyle 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N}\) ..*** EDIT *** Can now see that \(\displaystyle n^q \lt 1\) implies \(\displaystyle 1 / n^p \geq 1 / n \ \forall \ n \in \mathbb{N}\) ... indeed quite straightforward ... getting late here in Tasmania ...:(...
Sorry if I'm being a bit slow ...

Peter
 
Last edited:

1. What is the "P-series" in Sohrab, Proposition 2.3.12?

The "P-series" refers to a mathematical series of the form ∑(1/n^p), where n ranges from 1 to infinity and p is a constant exponent. This series is commonly used in calculus and number theory.

2. Who is Sohrab in Proposition 2.3.12?

Sohrab is the author of the mathematical text that contains Proposition 2.3.12. The full name of the author is not provided, so it is likely that Sohrab is a pseudonym.

3. What is Proposition 2.3.12 about?

Proposition 2.3.12 is a specific proposition within Sohrab's text. Without more context, it is impossible to determine the exact topic of this proposition.

4. What is the significance of Proposition 2.3.12?

The significance of Proposition 2.3.12 depends on its context within Sohrab's text. It may be a fundamental theorem or a key result that is used to prove other theorems.

5. How can Proposition 2.3.12 be applied in real-world situations?

Without knowing the topic of Proposition 2.3.12, it is difficult to determine its real-world applications. However, many mathematical concepts and theorems have practical applications in fields such as engineering, physics, and economics.

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