Strategy for Series — Question 10

*Question 10 Determine whether $\displaystyle\sum_{n=1}^{\infty}\frac{3n^2+1}{n^3-2}$ converges or diverges. Identify the dominant powers and use Limit Comparison with an appropriate benchmark series.

*Question 10 – Solution

The dominant-power quotient is $$\frac{3n^2}{n^3}=\frac3n,$$ so compare with the harmonic term $b_n=1/n$. A finite number of initial terms, including the negative $n=1$ term, does not affect convergence.

For $n\ge2$ both sequences are positive, and $$\begin{align*} \lim_{n\to\infty}\frac{a_n}{b_n} &=\lim_{n\to\infty}\frac{3n^2+1}{n^3-2}\cdot n\\ &=\lim_{n\to\infty}\frac{3+1/n^2}{1-2/n^3}=3. \end{align*}$$ The limit is finite and strictly positive.

Since $\sum1/n$ diverges, the given series diverges by the Limit Comparison Test. Dominant powers correctly predicted harmonic-size terms.