Convergence of series: 18 flashcards
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1. What is the necessary condition for convergence of a series \(\sum a_n\)?
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The terms must approach zero: \(\lim_{n \to \infty} a_n = 0\). This is necessary but not sufficient.
2. State the Geometric Series Test and its convergence condition.
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A geometric series \(\sum_{n=0}^{\infty} ar^n\) converges if \(|r| < 1\), with sum \(\frac{a}{1-r}\). It diverges if \(|r| \ge 1\).
3. For what values of \(p\) does the p-series \(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converge?
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The p-series converges if \(p > 1\) and diverges if \(p \le 1\).
4. Apply the Divergence Test to \(\sum_{n=1}^{\infty} \frac{n}{2n+1}\). Does it converge or diverge?
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Diverge, because \(\lim_{n \to \infty} \frac{n}{2n+1} = \frac{1}{2} \neq 0\).
5. When can the Integral Test be used, and what is the conclusion?
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If \(f(x)\) is positive, continuous, and decreasing for \(x \ge 1\) and \(a_n = f(n)\), then \(\sum a_n\) converges iff \(\int_1^\infty f(x)\,dx\) converges.
6. Use the Integral Test to determine convergence of \(\sum_{n=1}^{\infty} \frac{1}{n^2}\).
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Converges, because \(\int_1^\infty \frac{1}{x^2}\,dx = 1\) (finite).
7. State the Direct Comparison Test.
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If \(0 \le a_n \le b_n\) for all \(n\), then: (1) If \(\sum b_n\) converges, so does \(\sum a_n\). (2) If \(\sum a_n\) diverges, so does \(\sum b_n\).
8. Determine convergence of \(\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}\) using Direct Comparison.
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Converges, since \(\frac{1}{n^2+1} < \frac{1}{n^2}\) and \(\sum \frac{1}{n^2}\) converges.
9. State the Limit Comparison Test.
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If \(a_n > 0\), \(b_n > 0\), and \(\lim_{n \to \infty} \frac{a_n}{b_n} = c\) with \(0 < c < \infty\), then both series \(\sum a_n\) and \(\sum b_n\) either both converge or both diverge.
10. Use Limit Comparison to test \(\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3 + 1}}\).
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Converges. Compare with \(b_n = \frac{1}{n^{3/2}}\). The limit \(\lim \frac{a_n}{b_n} = 1\), and \(\sum \frac{1}{n^{3/2}}\) converges (p-series with p=1.5>1).
11. State the Ratio Test and its conclusion.
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For \(\sum a_n\) with \(a_n \neq 0\), let \(L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|\). If \(L < 1\), series converges absolutely; if \(L > 1\) or infinite, diverges; if \(L = 1\), test is inconclusive.
12. Apply the Ratio Test to \(\sum_{n=1}^{\infty} \frac{n}{2^n}\).
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Converges, because \(L = \lim \frac{(n+1)/2^{n+1}}{n/2^n} = \frac{1}{2} < 1\).
13. State the Root Test and its conclusion.
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For \(\sum a_n\) with \(a_n \ge 0\), let \(L = \lim_{n \to \infty} \sqrt[n]{a_n}\). If \(L < 1\), series converges; if \(L > 1\), diverges; if \(L = 1\), inconclusive.
14. Apply the Root Test to \(\sum_{n=1}^{\infty} \left(\frac{2n+1}{3n-1}\right)^n\).
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Converges, because \(\lim \sqrt[n]{a_n} = \lim \frac{2n+1}{3n-1} = \frac{2}{3} < 1\).
15. What is the Alternating Series Test (Leibniz's test)?
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For \(\sum (-1)^{n} b_n\) with \(b_n > 0\), if \(b_n\) is decreasing and \(\lim_{n \to \infty} b_n = 0\), then the series converges.
16. Determine if \(\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}\) converges.
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Converges by the Alternating Series Test, since \(b_n = \frac{1}{n}\) is decreasing and tends to 0. (It is the alternating harmonic series.)
17. What is the difference between absolute and conditional convergence?
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A series \(\sum a_n\) converges absolutely if \(\sum |a_n|\) converges. It converges conditionally if \(\sum a_n\) converges but \(\sum |a_n|\) diverges.
18. Classify \(\sum_{n=1}^{\infty} \frac{\sin n}{n^2}\) as absolutely convergent, conditionally convergent, or divergent.
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Absolutely convergent, because \(\left|\frac{\sin n}{n^2}\right| \le \frac{1}{n^2}\) and \(\sum \frac{1}{n^2}\) converges.
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