Determine whether the following sequences converge or...

Determine whether the following sequences converge or diverge and describe whether they do so monotonically or by oscillation. Give the limit when the sequence converges.
$$\left\{2^{n} 3^{-n}\right\}$$

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Key Concepts

Geometric Sequences

A geometric sequence is one in which each term is obtained by multiplying the previous term by a constant factor, known as the common ratio. The behavior of the sequence, including whether it converges or diverges, is determined by the absolute value of this ratio.

Convergence of Sequences

A sequence converges if its terms approach a specific finite value as the index goes to infinity. For geometric sequences, if the absolute value of the common ratio is less than one, the sequence converges; otherwise, it diverges.

Monotonic Sequences

A sequence is monotonic if its terms are either entirely non-increasing or non-decreasing throughout. In the context of a converging geometric sequence with a positive common ratio less than one, the sequence is monotonically decreasing.

Limits of Sequences

The limit of a sequence is the value that the terms of the sequence approach as the index becomes infinitely large. For a geometric sequence with an absolute common ratio less than one, the limit is zero.

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