Tutorial Week 11

Comparison Test

The comparison test states:

Given series \(\sum a_n\) and \(\sum b_n\), where \(a_n \ge 0\) and \(b_n \ge 0\),

1. If \(\sum a_n\) converges and \(b_n \le a_n\), then \(\sum b_n\) converges.

2. If \(\sum a_n\) diverges and \(a_n \le b_n\), then \(\sum b_n\) diverges.

Q1: Use the comparison test to show whether \(\sum_{n=1}^\infty \frac{3n + 5}{2^n + 5^n - 1}\) converges.

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Alternating Series Test

The alternating series test states:

Given an alternating series in the form of \(\sum (-1)^n b_n\) where \(b_n \gt 0\), if \(b_n\) forms a decreasing sequence (\(b_{n+1} \le b_n\)) and \(\lim_{n \to \infty} b_n = 0\), then \(\sum (-1)^n b_n\) converges.

Q2: Does the series \(\sum_{n=1}^\infty \frac{(-1)^nn^2}{n^2+n}\) converge? If it converges, is it absolutely or conditionally convergent?

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Q3: Does the series \(\sum_{n=1}^\infty \frac{(-1)^nn}{n^2+n}\) converge? If it converges, is it absolutely or conditionally convergent?

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