Sequence Limit Proof

Maburo · 2014-10-06 18:23:43

Hello. I am trying to prove the limit of a sequence using the definition of a sequence limit. The definition is:

I am trying to prove that

This is what I have done so far:

The issue I am having is that by the definition, epsilon can be any positive number. But by the value I have given N through manipulating these equations, epsilon cannot be equal to 2. Is this allowed? If not, is there some other method of finding such an N?

I appreciate any input! Thanks.

Olinguito · 2014-10-07 01:39:00

This step

Maburo wrote:

is not valid. Epsilon might be less than or equal to 2.

I don’t really know how you could go about choosing a suitable N for this problem. Let me think about it.

Maburo · 2014-10-07 15:33:30

Is there some way to define N as the minimum of two values (one being the value I already found), that would make this possible? I am completely stuck on this one. I can't figure out where to go with it

Olinguito · 2014-10-08 01:01:50

If you define N as the minimum of two values, both values must be positive. However for limit problems it’s more usual to define N as the maximum of two values, not minimum.

For problems of this kind you often have to be imaginative. And for this problem I think the key is this:

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Hence:

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Now for r = 1, …, n, we have

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Thus:

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So given you can take .

Maburo · 2014-10-08 02:25:16

Very clever! That was awesome. Now I suppose I should practice coming up with such things on spot.

Thanks for your help!

Olinguito · 2014-10-08 04:13:33

You’re welcome.

As I said, problems of this kind often demand a lot of ingenuity from the solver, but practice and experience can help a lot. Don’t worry if you can’t come up with such things on the spot – I certainly didn’t for this one! I had to think hard about it for a while. While thinking about it, I realized that the 1 was going to be a problem if I was going to use the triangle inequality, so I tried splitting the 1 into the terms of the sum – and it worked. I mean, it might not have worked and then I would have to start again from the beginning, but fortunately it did. So yeah – practice, practice, practice.

Math Is Fun Forum

#1 2014-10-06 18:23:43

Sequence Limit Proof

#2 2014-10-07 01:39:00

Re: Sequence Limit Proof

#3 2014-10-07 15:33:30

Re: Sequence Limit Proof

#4 2014-10-08 01:01:50

Re: Sequence Limit Proof

#5 2014-10-08 02:25:16

Re: Sequence Limit Proof

#6 2014-10-08 04:13:33

Re: Sequence Limit Proof

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