Answer:
divergent
Explanation:
We can use the root test.
The root test states that if the limit of the nth root of the absolute value of the nth term of a series is less than 1, then the series is convergent. If the limit is greater than or equal to 1, then the series is divergent. If the limit is indeterminate, then the root test is inconclusive and we need to use a different test.
To apply the root test to the series √ 1/4 +√ 2/6 + √ 3/8 + ........, we can take the nth root of the absolute value of the nth term of the series for each value of n. In this case, the nth term of the series is √ (n-1)/(2n). Therefore, we can take the nth root of the absolute value of this expression for each value of n:
√|√ (1-1)/(21)| = 1
√|√ (2-1)/(22)| = 1/√2
√|√ (3-1)/(23)| = 1/√3
√|√ (4-1)/(24)| = 1/2
As we can see, the limit of the nth root of the absolute value of the nth term of the series is 1, which is greater than or equal to 1. Therefore, according to the root test, the series √ 1/4 +√ 2/6 + √ 3/8 + ........ is divergent.
I hope this helps! Let me know if you have any other questions or need further assistance.