Comparison and Limit Comparison Tests — Question 1

*Question 1 Consider $\displaystyle\sum_{n=1}^{\infty}\frac1{n^2+3n}$.

1. Identify the dominant power and select a standard benchmark.

2. Prove a direct-comparison inequality with the correct direction.

3. Classify the series and confirm the result by limit comparison.

State why the benchmark converges or diverges.

*Question 1 – Solution

The denominator behaves like $n^2$, so use $b_n=1/n^2$, a convergent $p$-series with $p=2$.

Since $n^2+3n\ge n^2>0$, $$0<\frac1{n^2+3n}\le\frac1{n^2}.$$ The Direct Comparison Test proves convergence.

$$\lim_{n\to\infty}\frac{1/(n^2+3n)}{1/n^2} =\lim_{n\to\infty}\frac1{1+3/n}=1.$$ The finite positive limit confirms that both series share the same verdict.