Answer:
The sequence converges. The limit DNE.
Explanation:
Find the limit of n as n tends to infinity (in other words, positive infinity) in {Ln(n)/ Ln(3n)}
Positive infinity values for n start from 1,2,3,4,5,6,7,8,9,10,11,12,13,14,...,infinity
So I solved for values of n, up to n=20. All values are rounded up to 3 decimal places; for better accuracy.
When n is 1, the function is equal to 0.000
When n is 2, the function is = 0.387
When n is 3, the function = 0.500
When n is 4, the function = 0.558
When n is 5, the function = 0.594
When n is 6, the function = 0.619
When n is 7, the function = 0.639
When n is 8, the function = 0.654
When n is 9, the function = 0.667
When n is 10, the function = 0.677
When n is 11, the function = 0.686
When n is 12, the function = 0.693
When n is 13, the function = 0.700
When n is 14, the function = 0.706
When n is 15, the function = 0.711
When n is 16, the function = 0.716
When n is 17, the function = 0.721
When n is 18, the function = 0.725
When n is 19, the function = 0.728
When n is 20, the function = 0.732
We say there is a convergence because the space between the values of n gets smaller and smaller as n tends to infinity and there is no definite limit. Limit DNE.