To determine whether the given series

$\sum_{n=1}^\infty \frac{(-1)^n + 3n - 1}{n^2 + n}$

converges or diverges, we analyze the terms step by step.

Step 1: Examine the general term

The general term of the series is:

$a_n = \frac{(-1)^n + 3n - 1}{n^2 + n}.$

Here, $(-1)^n$ introduces an alternating component, while $3n - 1$ grows linearly as $n$ increases.

Step 2: Check the dominant term as $n \to \infty$

As $n$ becomes large, the denominator $n^2 + n$ grows quadratically, while the numerator grows linearly due to the $3n$ term. Thus, the general term $a_n$ behaves like:

$a_n \sim \frac{3n}{n^2 + n} = \frac{3}{n}.$

This suggests the series may behave similarly to the harmonic series, which diverges.

Step 3: Apply the divergence test

The divergence test (or $n$-th term test) states that if $\lim_{n \to \infty} a_n \neq 0$, the series diverges. Let's compute the limit of $a_n$:

$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{(-1)^n + 3n - 1}{n^2 + n}.$

Focus on the dominant terms in the numerator and denominator. As $n \to \infty$, the $3n$ term in the numerator dominates $(-1)^n$ and $-1$, and the $n^2$ term in the denominator dominates $n$. Thus:

$a_n \sim \frac{3n}{n^2} = \frac{3}{n}.$

Since $\frac{3}{n} \to 0$ as $n \to \infty$, the divergence test is inconclusive. We proceed to further tests.

Step 4: Analyze the series for convergence

The series is neither purely alternating nor purely positive, so direct tests for alternating or positive series do not apply straightforwardly. However, due to the growth rate of the terms, we suspect divergence. The comparison test or limit comparison test with a divergent series such as the harmonic series $\sum \frac{1}{n}$ might be useful.

Comparison with $\sum \frac{1}{n}$:

$a_n = \frac{(-1)^n + 3n - 1}{n^2 + n} \geq \frac{3n}{n^2 + n} = \frac{3}{n + 1}.$

Since $\sum \frac{3}{n + 1}$ diverges (as it behaves like a harmonic series), the original series also diverges by comparison.

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Conclusion:

The series $\sum_{n=1}^\infty \frac{(-1)^n + 3n - 1}{n^2 + n}$ diverges.

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Do you want a detailed explanation of any of these steps?

Related Questions:

1. How is the divergence test applied in other contexts?
2. What is the comparison test and when is it used?
3. How does alternating behavior affect convergence tests?
4. Why does the harmonic series diverge?
5. What are some examples of series that look similar but converge?

Tip:

Always simplify the dominant terms in the numerator and denominator to approximate the behavior of the series at infinity.