Comparison and Limit Comparison Tests — Question 7

*Question 7 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{3^n+1}{5^n-2}$.

1. Select a geometric benchmark from the dominant exponential terms.

2. Compute the Limit Comparison Test ratio.

3. Classify the series and identify the benchmark’s common ratio.

*Question 7 – Solution

The dominant quotient is $3^n/5^n=(3/5)^n$. Let $b_n=(3/5)^n$, whose common ratio $3/5$ has absolute value less than $1$; hence $\sum b_n$ converges.

$$\frac{(3^n+1)/(5^n-2)}{(3/5)^n} =\frac{1+3^{-n}}{1-2\cdot5^{-n}}\longrightarrow1.$$ The original terms and benchmark are positive for $n\ge1$, and the ratio limit is finite and positive.

The two series have the same convergence behavior. Since the benchmark is geometric and convergent, $$\boxed{\sum_{n=1}^{\infty}\frac{3^n+1}{5^n-2}\text{ converges}.}$$