48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.
∑ (k = 1 to ∞) 3ᵏ⁺² / 5ᵏ

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \frac{3^{k+2}}{5^k} \). Notice that the terms involve exponential expressions with base 3 and 5.
2. Rewrite the general term to simplify the expression: \( \frac{3^{k+2}}{5^k} = 3^2 \cdot \frac{3^k}{5^k} = 9 \cdot \left( \frac{3}{5} \right)^k \). This shows the series is a constant multiple of a geometric series.
3. Recognize that the series is geometric with common ratio \( r = \frac{3}{5} \). Recall that a geometric series \( \sum ar^k \) converges if and only if \( |r| < 1 \).
4. Since \( \left| \frac{3}{5} \right| < 1 \), the geometric series converges. Therefore, the original series converges as well.
5. To find the sum (if needed), use the formula for the sum of a geometric series starting at \( k=1 \): \[ S = a \cdot \frac{r}{1-r} \], where \( a = 9 \) and \( r = \frac{3}{5} \).

Key Concepts

Geometric Series

A geometric series is a series where each term is obtained by multiplying the previous term by a constant ratio. It has the form ∑ ar^k, and it converges if the absolute value of the ratio |r| < 1. Understanding this helps identify if the given series fits this pattern and whether it converges.

Convergence Tests for Series

Convergence tests, such as the geometric series test, ratio test, and root test, help determine if an infinite series converges or diverges. Applying these tests involves analyzing the behavior of terms as k approaches infinity to conclude about the sum's finiteness.

Manipulating Series Terms

Rewriting series terms into a recognizable form, such as factoring constants or expressing terms with exponents clearly, is essential. This simplification allows easier application of convergence tests and better insight into the series' structure.