Question

Determining Convergence or DivergenceWhich of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

Accepted Answer

First, examine the general term of the series: $$a_n = \frac{2^n + 3^n}{3^n + 4^n}. To $$determine convergence or divergence, analyze the behavior of $$a_n as$$ $$n$$ approaches infinity. Identify the dominant terms in the numerator and denominator for large $$n. $$Since exponential functions grow at different rates, compare $$3^n$$ and $$4^n in $$the denominator, and $$2^n$$ and $$3^n in $$the numerator. Simplify the term by dividing numerator and denominator by the largest exponential term in the denominator, which is $$4^n. $$This gives: $$a_n = \frac{\frac{2^n}{4^n} + \frac{3^n}{4^n}}{\frac{3^n}{4^n} + 1}. $$Rewrite the fractions inside the numerator and denominator using properties of exponents: $$\left(\frac{2}{4}ight)^n$$, $$\left(\frac{3}{4}ight)^n$$, and $$\left(\frac{3}{4}ight)^n. $$Analyze the limit of $$a_n as$$ $$n 	o \infty by $$considering these terms. If $$\lim_{n 	o \infty} a_n 
eq 0$$, then by the Test for Divergence, the series diverges. If the limit is zero, consider applying other convergence tests such as the Comparison Test or Ratio Test to determine if the series converges.

Determining Convergence or Divergence
Which of the series in Exercises 17–56 converge, and which diverge? Use any method, and give reasons for your answers.
∑ (from n=1 to ∞) (2ⁿ + 3ⁿ) / (3ⁿ + 4ⁿ)

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Key Concepts

Infinite Series and Convergence

Limit Comparison Test

Behavior of Exponential Terms in Series