Question

If you have the sum nᵏ ⋅ rⁿ when does it converge?

a) r < 1

b) r > 1

c) k < 0

d) n → [infinity]

Answer 1

The sum converges when r is less than 1 and as n approaches infinity. If r is greater or equal to 1, the terms do not approach zero, leading to divergence.

Step-by-step explanation:

If you have the sum nᵗ · rⁿ, the convergence of this series depends on the value of r. Specifically, this sum will converge under certain conditions which are related to the values of r and the limit as n approaches infinity.

Convergence Conditions

The series converges when r < 1. It is essential to consider the behavior of the sequence as n approaches infinity (n → ∞). When r is less than 1, as n increases, the terms of the series become smaller and smaller, ultimately approaching zero, which allows the series to converge to a finite sum. However, if r >= 1, the terms do not approach zero, and hence the series will not converge but instead diverge or grow without bound.