How to prove limn / A ^ n's limit is 0 when n tends to infinity by using limit existence criterion

Not necessarily, if 0

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1. It is proved that: (n - > infinity) limn ^ (1 / N) = 1
2. Finding the limit of LiMn →∞ ((3 ^ n + 2 ^ n) / (3 ^ (n + 1) - 2 ^ (n + 1))) Ask for detailed explanation
3. LIM (x ^ n) - 1 / (x-1) n is a positive integer x → 1 1 + (the third power of U) / (1 + U) x → positive infinity under the root sign of lim4 Lim-x → positive infinity
4. The limit of LIM (x ^ n-1) / (x-1) (x tends to 1n is a positive integer)
5. Lim n ^ 2 {(K / N) - (1 / N + 1) - (1 / N + 2) - (1 / N + k)} (where k is a constant independent of n)
6. Verification: Lim1 ^ k + 2 ^ k + 3 ^ k + 4 ^ k +. N ^ k / N ^ (K + 1) = 1 / K + 1 N is a positive integer followed by K + 1
7. LIM (n →∞) (n ^ 2 + 2) / N + kn = 0 for K
8. The detailed process of finding LIM (n →∞) ∑ (k = 1, n) K / (n ^ 2 + N + k) limit
9. (n - > 00) LIM (n + k) / (n ^ 2 + k) (n from 1 to n)
10. Find Lim n → + ∞ (1 / N ^ k + 2 / N ^ k + + n / N ^ k) There are three situations,
11. LiMn →∞ (1 + 2 + 3 +...) n-1)/n²
12. limn^(1/n) n-->∞=? N is not a constant
13. Limit: LiMn →∞ (n-1) ^ 2 / (n + 1)
14. Finding limit LiMn →∞ (n-1) ^ 2 / (n + 1) Limit: LiMn →∞ (n-1) ^ 2 / (n + 1) Is to find the limit of the square of (n-1) divided by (n + 1) when n approaches infinity I think we can use that formula, that is, n tends to infinity, and the limit is determined by the degree of N, because the upper side is the square of N, and the lower side is the first power of N, so I think it should be ∞ But the answer is + ∞. How does that work out? The upper part of the fractional line should be approaching positive infinity. Will the positive infinity of the answer come like this This is not a series limit problem, just a common function limit problem, there should be no statement that n must approach positive infinity, right?
15. LIM (n →∞) un * n = 0, then the series ∑ UN converges?
16. Find the convergence and divergence of a series. LIM (n tends to infinity) is one of the nth power of the sum of 1 + 1 / N. find the convergence and divergence of this series
17. Prove Lim n - > infinite n ^ n / (n!) ^ 2 = 0 by using necessary conditions of series convergence Please
18. Prove Lim n →∞ n ^ n / (n!) ^ 2 = 0 by using necessary conditions of series convergence
19. Let LIM (n →∞) na_ N exists and the series ∑ (n = 1 →∞) n (a)_ n-a_ It is proved that the series ∑ (n = 1 →∞) a is convergent_ N convergence
20. Let the positive series ∑ UN and ∑ VN converge. It is proved that ∑ (UN + VN) ^ 2 also converges ……