Rudin's Proof of Lim n->∞ (p^(1/n)) = 1

In summary, Rudin uses the binomial theorem to prove lim n-> inf (p^(1/n)) = 1 by showing that the first two terms of the expansion are less than or equal to the whole expression. He also uses the binomial theorem to show that one term of the expansion is greater than or equal to the whole expression.
  • #1
samspotting
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Rudin's proof of lim n-> inf (p^(1/n)) = 1

1+n*x_n <= (1 + x_n)^n = o

I don't see it from the binomial theorem, which is what he says that is from.

He also does things with the binomial theorem like:

(1+x_n)^n >= ((n(n-1)) / 2) *x_n^2

I'm not sure what he did to get these two inequalities.
 
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  • #2
Those are terms in the expansion given by the binomial theorem. Since there are more terms in the actual expansion, it's an inequality.
 
  • #3
samspotting said:
Rudin's proof of lim n-> inf (p^(1/n)) = 1

1+n*x_n <= (1 + x_n)^n = o
1+ n*x_n <= (1+ x_n)^n- o perhaps, not "= o". (1+ x_n)^n, by the binomial theorem that you mention in your title, says that (1+ x_n)^n= 1+ n x_n+ terms of higher order in x_n which will go to 0 as x_n goes to 0: o(x_n) or "small o". Assuming that x_n is positive, all those missing terms in the binomial expansion are positive and so the first two terms are less than or equal to the whole thing.

I don't see it from the binomial theorem, which is what he says that is from.

He also does things with the binomial theorem like:

(1+x_n)^n >= ((n(n-1)) / 2) *x_n^2

I'm not sure what he did to get these two inequalities.
n(n-1)/2 is the second binomial coefficient nC2= n!(2!(n-2)!)= n(n-1)/2. (1+ x_n)^n= 1+ n x_n+ (n(n-2)/2) x_n^2+ higher terms. Again, assuming that x_n is positive, the 'whole thing' is greater than or equal to just one term.
 
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  • #4
Thanks. I feel pretty stupid lol, but its been a while since I've seen that thing.
 

FAQ: Rudin's Proof of Lim n->∞ (p^(1/n)) = 1

What is Rudin's Proof of Lim n->∞ (p^(1/n)) = 1?

Rudin's Proof of Lim n->∞ (p^(1/n)) = 1 is a mathematical proof that shows the limit of the expression p^(1/n) is equal to 1 as n approaches infinity. This proof was first presented by mathematician Walter Rudin in his book "Principles of Mathematical Analysis".

Why is this proof important?

This proof is important because it helps to establish the concept of limits in mathematics. It also demonstrates the relationship between exponential and logarithmic functions, and how they behave as n approaches infinity.

What is the significance of the value 1 in this proof?

The value 1 in this proof represents the idea of convergence. It shows that as n approaches infinity, the expression p^(1/n) approaches a single value, which is 1. This concept is important in understanding the behavior of functions and sequences in mathematics.

How does Rudin's Proof of Lim n->∞ (p^(1/n)) = 1 relate to real-world applications?

While this proof may not have direct real-world applications, the concepts and techniques used in the proof are important in various fields such as physics, engineering, and economics. Understanding limits and their properties is essential in modeling and analyzing real-world phenomena.

Are there any limitations to this proof?

Like any mathematical proof, there may be limitations to Rudin's Proof of Lim n->∞ (p^(1/n)) = 1. This proof relies on certain assumptions and techniques, and may not be applicable in all situations. It is important to carefully analyze the assumptions and conditions under which this proof holds true.

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