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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.
"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein
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This step
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.
Bassaricyon neblina
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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
"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein
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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:
[list=*]
[*]
Hence:
[list=*]
[*]
Now for r = 1, …, n, we have
[list=*]
[*]
Thus:
[list=*]
[*]
Bassaricyon neblina
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Very clever! That was awesome. Now I suppose I should practice coming up with such things on spot.
Thanks for your help!
"Pure mathematics is, in its way, the poetry of logical ideas."
-Albert Einstein
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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.
Bassaricyon neblina
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