(Solved): Does the following series converge or diverge? n=16n+73n3 The series diverges by ...

Expert Answer

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To determine whether the series
  

converges or diverges, we can examine the different convergence tests. Let's analyze each option:

The series diverges by the Limit Comparison Test: The Limit Comparison Test states that if series A and series B are such that the limit of the ratio A(n)/B(n) as n approaches infinity is a positive finite value, then both series either converge or diverge together. In this case, we would need to compare the given series to another series to apply this test. Since no other series is mentioned, we cannot conclude that the series diverges by the Limit Comparison Test.
The series diverges by the Root Test: The Root Test states that if the nth root of the absolute value of the terms of the series approaches a value greater than 1 as n approaches infinity, then the series diverges. In this case, we need to compute the nth root of the absolute value of the terms of the series and check if it is greater than 1. Let's calculate it:
  
To simplify the calculation, let's multiply the numerator and denominator by the conjugate of the denominator:
  
  
Simplifying further, we get:
  
Now, as n approaches infinity, the term (7 * sqrt(3) * n) dominates over the constant term (6sqrt(6)). Therefore, we can simplify the expression to:
  
  
Simplifying further, we have:
  
  
Since both 2   are positive constants, the limit as n approaches infinity is equal to 1. Therefore, the nth root of the absolute value of the terms of the series approaches 1, and we cannot conclude that the series diverges by the Root Test.