Tutorial Week 9

Limits, convergence, and divergence of Sequences

Q1: Given the sequence \(\{ a_n \}\) where \(a_n = \frac{(n + n^2)^{\frac{1}{3}}}{n}\), find if it converges or diverges. If it converges, find the limit.

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Q2: Given the sequence \(\{ a_n \}\) where \(a_n = sin(n) \frac{log_3(n)}{log_8(n)}\), find if it converges or diverges. If it converges, find the limit.

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Monotone Convergence Theorem (MCT) and Convergence

Q3: Show that the sequence \(a_n = \frac{2n-1}{3n+4}\) is monotonic.

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Q4: Show how you can use MCT to prove that the sequence \(\{ \frac{n}{e^n} + 1 \}\) is convergent.

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Limits of Monotonic Sequences

Q5: Find the limit of the convergent sequence \(\{ \sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, ... \}\).

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Q6: Show that the sequence \(a_{n+1} = 1 + \frac{1}{1+a_n}\) converges to \(\sqrt{2}\).

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