Princeton Review Typo Page 77

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Legendre
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Princeton Review Typo Page 77

Postby Legendre » Sat Aug 22, 2009 9:50 am

3rd edition. Solution to example 2.27

"An = f(n) then sequence (An) converges to L <=> f(x) converges to L as x -> infinity"


I think the statement is wrong in that (An) converges to L do not => f(x) converges to L as x -> infinity.

Suppose the statement is true.

Counter example :

Let f(x) be a function s.t. f(x) = M for all x that are no natural numbers. Where M not equal L. And f(x) = Ax for x that are natural numbers. So this function satisfies An = f(n) for all natural numbers n but has the value M for all inputs that are non-natural numbers.

Although (An) converges to L, this function f do not as x -> infinity.

Nameless
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Re: Princeton Review Typo Page 77

Postby Nameless » Thu Sep 03, 2009 10:27 pm

An = f(n) then sequence (An) converges to L <=> f(x) converges to L as x -> infinity"

unless I am misunderstanding some thing, the above statement is true. Your example does not work

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lime
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Re: Princeton Review Typo Page 77

Postby lime » Fri Sep 04, 2009 1:21 am

The function f(x) must be constructed just by "substituting" "x" instead of "n" in formula for f(n). Not in the arbitrary way as you did.

Legendre
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Re: Princeton Review Typo Page 77

Postby Legendre » Sat Sep 05, 2009 8:27 am

Thanks a lot for the input guys, I will ponder over this.

Legendre
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Re: Princeton Review Typo Page 77

Postby Legendre » Sat Sep 05, 2009 8:48 am

lime wrote:The function f(x) must be constructed just by "substituting" "x" instead of "n" in formula for f(n). Not in the arbitrary way as you did.



Are you saying that the domain of function f(x) must be the set of natural numbers?

If f : R -> R then why can't it be true that An = f(n) for all natural numbers n but f(n) = a constant value for all real numbers that are not natural numbers?


*It is not stated that (An) = f(n) only for all natural numbers n. If I assume it is only true for all natural numbers n, then I am right?

Legendre
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Re: Princeton Review Typo Page 77

Postby Legendre » Fri Aug 24, 2012 6:00 am

I was right: if the function converges, then the sequence converges. (the function converges to the same limit even if the input sequence is restricted to those in the original sequence)

But not the other way around.




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