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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 (since for all ) that So then we have, Thus, if then as well (since ). So, for every we have . We compute for the given values of as follows:

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

1. Artem says:

The answer is not correct. You are dividing by FACTORIAL, so the values of N are actually 2, 4, 5, 7, 8. Just compute the value for N = 5: 5! = 120. 1/ 5! 1 / 120 < 0.01! Which corresponds to epsilon = 0.01! Thus the answer N = 101 is incorrect.

2. *Here the values of N shoud be 2,11,101,1001,10001. As according to the question.Please correct me if Iam wrong.

• RoRi says:

I don’t understand where you get those values of $N$. I think the above is correct, but maybe I have made a mistake somewhere?