## Princeton Review Typo Page 77

Forum for the GRE subject test in mathematics.
Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

### Princeton Review Typo Page 77

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
Posts: 128
Joined: Sun Aug 31, 2008 4:42 pm

### Re: Princeton Review Typo Page 77

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

lime
Posts: 129
Joined: Tue Dec 04, 2007 2:11 am

### Re: Princeton Review Typo Page 77

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
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

### Re: Princeton Review Typo Page 77

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

Legendre
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

### Re: Princeton Review Typo Page 77

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
Posts: 217
Joined: Wed Jun 03, 2009 1:05 am

### Re: Princeton Review Typo Page 77

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.