Question

Determine whether the series is convergent or divergent.
∑ₙ=1[infinity]5+(-1)ⁿ/n √(n)

Answer 1

To determine whether the given series is convergent or divergent, we can use the limit comparison test and compare it to a known convergent series. Taking the limit as n approaches infinity, the given series is found to be divergent.

Step-by-step explanation:

To determine whether the series is convergent or divergent, we can use the limit comparison test. Let's compare it to the series ∑ₙ=1⟮∞⟯1/√n, which is a known convergent series. Taking the limit as n approaches infinity of the ratio of the terms of the given series and the known convergent series, we get:

lim (n→∞) [5+(-1)ⁿ/n√n] / [1/√n] = lim (n→∞) (5n√n + (-1)ⁿ) / n = lim (n→∞) (5√n + (-1)ⁿ/n) = ∞

Since the limit is not equal to 0, the given series does not converge. Therefore, it is divergent.