Hi, my questions are:
1)Circle the convergent sequences in the following list (note: I haven't included the whole list), and find the limit in case the sequence converges.
b) 0.9, 0.99, 0.999, ...
For 1a) : By calculating the first few terms, I found that as n tends to infinity, n^(1/n) tends to 1, so I know that limit is 1, but I need to show this formally using the definition of convergence: for all epsilon, e>0 there exists a natural number N such that for all n>N, |an - L|<e.
All I've done so far is say that |n^(1/n) - 1| , however I'm not sure which N to choose so that for all n>N, this absolute value will be less than epsilon e.
For 1b) I know that the sequence tends to 1 as n tends to infinity, however I'm not sure how to prove this formally. I thought of using the theorem that any bounded and monotonically increasing or decreasing sequence is convergent, and I know that I'd need to use proof by induction to show both, but again I'm not quite sure how to do this.
Any help would be greatly appreciated!