Question

Is Σ(-1)ⁿ ⋅ 1/√n convergent?

a) Absolutely convergent

b) Conditionally convergent

c) Divergent

d) Alternating series test

Answer 1

The series Σ(-1)ⁿ ⋅ 1/√n is conditionally convergent based on the Alternating Series Test, as the terms are positive and decreasing and their limit as n approaches infinity is zero.

Step-by-step explanation:

To determine whether the series Σ(-1)ⁿ ⋅ 1/√n is convergent, we can apply the Alternating Series Test (AST), which states that a series of the form Σ(-1)ⁿ ⋅ aₙ is convergent if two conditions are met:

1. The terms aₙ must be decreasing in absolute value.
2. The limit of aₙ as n approaches infinity must be zero.

For the given series, the terms are 1/√n, which are positive and decreasing. Thus, the first condition is met. The limit as n approaches infinity of 1/√n is zero, satisfying the second condition of the AST. Therefore, the series is conditionally convergent since it is convergent by the AST but not absolutely convergent (because Σ 1/√n is a p-series with p = 1/2, and such series are divergent for p ≤ 1).