Question

Test the series below for convergence using the Root Test:

∑[n=1 to [infinity]] ((8n²)/(9n+1))ⁿ

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

Answer 1

Using the Root Test to determine the convergence of the series, we find that the nth root of the nth power of the terms in the series leads to a limit of 8/9 when n approaches infinity. Since this limit is less than 1, the series converges.

Step-by-step explanation:

The student has asked to test the series ∑[n=1 to ∞] ((8n²)/(9n+1))ⁿ for convergence using the Root Test. To apply the Root Test, we need to compute the following limit:

lim[n→∞] √[n](((8n²)/(9n+1))ⁿ)

This limit simplifies to the limit of the nth root of the nth power of the terms in the series. The function f(n) for the Root Test in this series will be:

f(n) = (8n²)/(9n+1)

To use the Root Test, we look for:

L = lim[n→∞] (f(n))^(1/n)

If L < 1, the series converges; if L > 1, it diverges; and if L = 1, the test is inconclusive.

For the series at hand, calculating the limit as n approaches infinity of the nth root (which is essentially the nth power removed due to the nth root) of f(n), we would have:

L = lim[n→∞] (8n² / (9n+1))

As n approaches infinity, the ratio of n² to n in the denominator becomes negligible, and we would be left evaluating the limit of 8/9, which is less than 1, indicating that the series converges.