11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.
∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)

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Rewrite the general term of the series to simplify the expression. The term is given by \(\frac{3^{k+4}}{5^{k-2}}\). Use the properties of exponents to separate the powers of constants and the variable \(k\): \(\frac{3^{k} \cdot 3^{4}}{5^{k} \cdot 5^{-2}} = 3^{4} \cdot 5^{2} \cdot \frac{3^{k}}{5^{k}}\).

Simplify the term further by combining the powers with the same base in the numerator and denominator: \(3^{4} \cdot 5^{2} \cdot \left(\frac{3}{5}\right)^{k}\). This shows the term is a constant multiplied by a geometric term \(\left(\frac{3}{5}\right)^{k}\).

Recognize that the series is a geometric series of the form \(\sum_{k=0}^{\infty} ar^{k}\) where \(a = 3^{4} \cdot 5^{2}\) and \(r = \frac{3}{5}\). Recall that a geometric series converges if and only if \(|r| < 1\).

Check the value of the common ratio \(r = \frac{3}{5}\). Since \(\frac{3}{5} = 0.6\) and \(0.6 < 1\), the geometric series converges.

Conclude that the original series converges by the geometric series test because the common ratio's absolute value is less than 1.

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Infinite Series and Convergence

An infinite series is the sum of infinitely many terms. Determining convergence means checking if the sum approaches a finite limit as the number of terms grows indefinitely. Understanding the behavior of the series terms is essential to decide if the series converges or diverges.

Geometric Series

A geometric series has terms that are multiples of a constant ratio raised to increasing powers. It converges if the absolute value of the common ratio is less than one, and its sum can be found using a specific formula. Recognizing a series as geometric simplifies the convergence test.

Convergence Tests for Series

Various tests, such as the Ratio Test or Root Test, help determine if a series converges. These tests analyze the limit of the ratio or root of successive terms. Applying the appropriate test provides a rigorous justification for convergence or divergence.

11–86. Applying convergence tests Determine whether the following series converge. Justify your answers.∑ (from k = 0 to ∞) (3ᵏ⁺⁴) / (5ᵏ⁻²)