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

검증된 단계별 안내

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.

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.

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 \( |r| < 1 \).

Since \( \left| \frac{3}{5} \right| < 1 \), the geometric series converges. Therefore, the original series converges as well.

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 \( r = \frac{3}{5} \).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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