Does the series converge or diverge? Use the Root test
((1/n)-(1/n^2))^n
To determine whether the series ((1/n)-(1/n^2))^n converges or diverges, we can use the Root test.
The Root test states that if the sequence of the nth roots of the absolute values of the terms of a series converges to a number less than 1, then the series converges absolutely. If, however, the sequence diverges or converges to a number greater than 1, then the series diverges. If the sequence converges to exactly 1, the test is inconclusive and another test may be needed.
Let's apply the Root test to the series ((1/n)-(1/n^2))^n:
First, we take the absolute value of the terms of the series:
|((1/n)-(1/n^2))^n| = ((1/n)-(1/n^2))^n
Then, we take the nth root of the absolute value of each term:
Limit as n approaches infinity of ((1/n)-(1/n^2)) = 0
Since the limit of the nth root of the absolute value of the terms is 0, which is less than 1, we can conclude that the series converges absolutely.
Therefore, the series ((1/n)-(1/n^2))^n converges.