Question

Test the series below for convergence using the Root Test:

∑[n=1 to [infinity]] ((4n+1)/(2n+4))ⁿ

The limit of the root test simplifies to lim[n→[infinity]] , Where f(n) = ___

Answer 1

The limit for the Root Test of the given series is 2, which indicates that the series diverges.

Step-by-step explanation:

To test the series for convergence using the Root Test, we need to find the limit of the nth root of the nth term of the series as n approaches infinity. We start by finding the nth root of the given term:

((4n+1)/(2n+4))^n

Now, we take the nth root of this expression:

nth root of ((4n+1)/(2n+4))^n = (4n+1)/(2n+4)

Next, we find the limit as n approaches infinity:

lim[n→∞] (4n+1)/(2n+4)

As n approaches infinity, the largest terms in the numerator and denominator dominate, so we can simplify the limit:

lim[n→∞] (4n)/(2n) = 2

Since the limit is greater than 1, the Root Test indicates that the series diverges. Hence, the function f(n) that the student is looking for would be:

f(n) = (4n+1)/(2n+4)

But for the Root Test, we are interested in the limit of this function as n approaches infinity:

lim[n→∞] f(n) = 2