How to prove that the sequence an=n1/n is convergent using definition of convergence?
Answer
Noticing that n1n>1 for all n, it all comes down to showing that for any ϵ>0, there is a n such that (1+ϵ)≥n1n, or by rearranging, that
(1+ϵ)n≥n
Now, let's first of all choose an m such that (1+ϵ)m is some number bigger than 2, let's say the smallest number greater than 3 that you can get. From here, swap m for 2m. This will make the left side a little over 3 times larger, and the right side 2 times larger. The next doubling will still double the right side, but the left side will increase roughly 9-fold. Repeating, we can easily see that the left side will at some point overtake the right side, and we have our n
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