Question

Use Raabe’s Test to determine whether the series

converges.
∑ₙ=1[infinity]√(n !)/(2+√(1))(2+√(2)) …(2+√(n))

Answer 1

To determine whether the series converges, we can make use of Raabe's Test. Raabe's Test is inconclusive for this series.

Step-by-step explanation:

To determine whether the series converges, we can make use of Raabe's Test. Raabe's Test is used to test the convergence or divergence of a series of the form ∑(n=1 to infinity) a_n where a_n is a positive sequence.

First, let's find the limit of the ratio (n/(n+1)) * (2+√(n+1))/(2+√(n)) as n approaches infinity. Taking the limit gives us:

lim(n→∞) [(n/(n+1)) * (2+√(n+1))/(2+√(n))] = 1.

Since the limit is equal to 1, Raabe's Test is inconclusive. Hence, we cannot determine the convergence of the given series using Raabe's Test. We may need to use other tests such as the ratio test or the root test.