Study guide: Series with Arbitrary Signs

📋 Course Outline

1. Alternating Series and Their Terms
2. Alternating Series Test
3. Alternating Riemann Series
4. Absolute Convergence
5. Conditional Convergence
6. Decision Strategy for Signed Series

📖 1. Alternating Series and Their Terms

🔑 Key Concepts & Definitions

• Alternating series: A real series whose terms can be written as $u_n=(-1)^n a_n$ or $u_n=(-1)^{n+1}a_n$ for a positive sequence $a_n$ uniquely given by $a_n=|u_n|$.

📝 Essential Points

• In an alternating series, the signs of consecutive terms alternate, so even-indexed and odd-indexed terms have opposite signs.

📖 2. Alternating Series Test

★ Must-know

📌 The Alternating Series Test states that a series $u_n=(-1)^n a_n$ converges if $a_n\geq 0$ is eventually decreasing and $a_n\to0$.

Further detail

• For a convergent alternating series, the remainder after the nth partial sum satisfies $|R_n|\leq a_{n+1}=|u_{n+1}|$ and has the same sign as the first omitted term $u_{n+1}$.

• The sum $S$ of a convergent alternating series lies between consecutive partial sums: $S_{2n+1}\leq S\leq S_{2n}$.

• For $u_n=\frac{(-1)^n}{n+1}$, the magnitudes $a_n=\frac1{n+1}$ decrease to zero, so the series converges by the Alternating Series Test.

💡 Memory Hook

Decreasing magnitudes → zero → convergence

📖 3. Alternating Riemann Series

★ Must-know

📌 The alternating Riemann series $\sum_{n\geq1}\frac{(-1)^n}{n^\alpha}$ converges if and only if $\alpha>0$.

Further detail

📌 The standard Riemann series $\sum_{n\geq1}\frac1{n^\alpha}$ converges if and only if $\alpha>1$.

• For $\alpha\leq0$ in the alternating Riemann series, the term magnitude $n^{-\alpha}\geq1$ does not tend to zero, so the series diverges by the term test.

💡 Memory Hook

Alternating series converge for α > 0; positive p-series require α > 1

📖 4. Absolute Convergence

🔑 Key Concepts & Definitions

• Absolute convergence: A numerical series $\sum u_n$ is absolutely convergent if the series $\sum |u_n|$ converges.

★ Must-know

📌 If $\sum |u_n|$ converges, then $\sum u_n$ converges and $\left|\sum_{n=0}^{\infty}u_n\right|\leq\sum_{n=0}^{\infty}|u_n|$.

Further detail

• To test a series with arbitrary signs for absolute convergence, study $\sum |u_n|$ using positive-series tests such as the Riemann, comparison, ratio, or root test.

• Since $\left|\frac{\cos n}{n^2}\right|\leq\frac1{n^2}$ and $\sum 1/n^2$ converges, $\sum_{n\geq1}\frac{\cos n}{n^2}$ converges absolutely by comparison.

• For $u_n=\frac{(-1)^n n^3}{n!}$, the absolute-term ratio tends to zero, so the series of absolute values converges by the Ratio Test and the original series converges absolutely.

💡 Memory Hook

Convergence of absolute values → convergence of the original series

📖 5. Conditional Convergence

🔑 Key Concepts & Definitions

• Conditional convergence: A series is conditionally convergent (or semi-convergent) if it converges but its series of absolute values diverges.

📝 Essential Points

• The alternating harmonic series $\sum_{n\geq1}(-1)^n/n$ converges by the Alternating Series Test, but $\sum_{n\geq1}1/n$ diverges, so the alternating series is conditionally convergent.

• The series $\sum_{n\geq1}(-1)^n/\sqrt n$ converges by the Alternating Series Test, but $\sum_{n\geq1}1/\sqrt n$ diverges, so the alternating series is conditionally convergent.

💡 Memory Hook

Convergent, but not absolutely convergent

📖 6. Decision Strategy for Signed Series

★ Must-know

• To determine the nature of a series with arbitrary signs, first check whether $u_n\to0$; then test $\sum|u_n|$; if it diverges, check the Alternating Series Test, and otherwise use an asymptotic expansion of the general term.

Further detail

• For $u_n=(-1)^n(\sqrt{n+1}-\sqrt n)$, rationalization gives $|u_n|=1/(\sqrt{n+1}+\sqrt n)\sim1/(2\sqrt n)$, so the absolute-value series diverges while the magnitudes decrease to zero; hence the series is conditionally convergent.

💡 Memory Hook

Term limit → absolute convergence → AST → asymptotic expansion

📊 Synthesis Tables

Convergence Criteria for Riemann Series

Series	Convergence condition
Standard $\sum 1/n^\alpha$	$\alpha>1$
Alternating $\sum(-1)^n/n^\alpha$	$\alpha>0$