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# Find an N such that the convergent sequence is within ε of its limit

Consider the convergent sequence with terms defined by Let . Find the value of and values of such that for all for each of the following values of :

1. ,
2. ,
3. ,
4. ,
5. .

First, we know So then we have, Thus, if then for every we have . We compute for the given values of as follows:

1. implies .
2. implies .
3. implies .
4. implies .
5. implies .

1. tom says:

I find myself nitpicking here: When you passed to the absolute value you must have assumed the limit L is zero, which is obvious in this case. But beyond knowing the limit in advance Apostol didn’t give any tools for actually determining convergence. What little I know of analysis it seems something like all convergent subsequences sharing the same limit would work, or perhaps cauchy criterion? It just seems that Apostols exposition of sequences was less then comforting; but maybe it will be covered later.

• Anonymous says:

Greetings!

There are of course alternative to solve this exercise. You can take and show that for we indeed have for arbitrary. Thus proving as using only the definition of convergence. From you can also create the required table of . (I use instead of ).

• Anonymous says:

Yeah but it should be ?