Limits of sequences

• Aug 25th 2013, 06:58 PM
Kristen111111111111111111
Limits of sequences
How to prove that 1/2 can't be the limit of the sequence {(3n-2)/(4n+1)} ?

I have tried prove this by the negation of the definition that is there exist an epsilon for all n0 belongs to positive integers such that n> n0 implies |(3n-2)/(4n+1)-1/2| >= epsilon.

But the problem is could not find an epsilon such that |(3n-2)/(4n+1)-1/2| is greater than or equal to that epsilon.

Any help would be great..
• Aug 26th 2013, 02:15 AM
emakarov
Re: Limits of sequences
Quote:

Originally Posted by Kristen111111111111111111
I have tried prove this by the negation of the definition that is there exist an epsilon for all n0 belongs to positive integers such that n> n0 implies |(3n-2)/(4n+1)-1/2| >= epsilon.

If ∃ denotes "there exists" and ∀ denotes "for all", then your version of negation of the limit definition is

∃ε ∀n0 ∀n. n > n0 ⇒ |xn - 1/2| ≥ ε

where xn = (3n-2)/(4n+1). I wrote ∀n because you did not provide a quantifier for n, and by default the quantifier is universal. In fact, the correct negation is

∃ε ∀n0 ∃n. n > n0 and |xn - 1/2| ≥ ε.

In words, there exists an ε such that there exist arbitrarily far elements of the sequence whose distance to 1/2 is at least ε. One way to prove that there exist arbitrarily far elements of the sequence with some property P is to show that all elements possess this property starting from some point. That is, to prove

∀n0 ∃n. n > n0 and P(xn) (1)

we show

∃n0 ∀n. n > n0 ⇒ P(xn) (2)

because (2) is stronger that (1). (Why?)

To prove (2), note that xn approaches 3/4 from below. Therefore, eventually it is greater than say, the midpoint between 1/2 and 3/4, i.e., 5/8. So, take ε = 5/8 - 1/2 = 1/8 and find an n0 such that xn ≥ 5/8 for all n > n0.
• Aug 26th 2013, 03:08 AM
Prove It
Re: Limits of sequences
Quote:

Originally Posted by Kristen111111111111111111
How to prove that 1/2 can't be the limit of the sequence {(3n-2)/(4n+1)} ?

I have tried prove this by the negation of the definition that is there exist an epsilon for all n0 belongs to positive integers such that n> n0 implies |(3n-2)/(4n+1)-1/2| >= epsilon.

But the problem is could not find an epsilon such that |(3n-2)/(4n+1)-1/2| is greater than or equal to that epsilon.

Any help would be great..

The limit is 3/4. This is clearly not 1/2.
• Aug 26th 2013, 07:16 AM
Kristen111111111111111111
Re: Limits of sequences
Well the 2nd one is stronger because the 2nd one can imply the first one..
Thank you so much for your help :)