📝 Absolute convergence of series (37 MCQs)

📖 Explanation: This question targets a fundamental misconception regarding the relationship between absolute and conditional convergence. While it is true that absolute convergence implies convergence, the converse is false. A series may diverge absolutely (meaning the sum of magnitudes grows without bound) yet still converge conditionally due to cancellation effects between positive and negative terms, such as in the alternating harmonic series. Understanding this distinction is crucial for higher-order analysis of series behavior.

📖 Explanation: This scenario describes the precise definition of conditional convergence. The key conceptual understanding required here is distinguishing between the behavior of the original series and the series of absolute values. If the original series converges to a limit $L$ but the series formed by taking the absolute value of each term fails to converge (is unbounded), the series relies on the specific ordering and sign alternation of terms to achieve its sum. This makes it conditionally convergent rather than absolutely convergent or purely divergent.

📖 Explanation: In physical modeling and advanced mathematics, the Riemann Rearrangement Theorem states that conditionally convergent series can be rearranged to sum to any real number or diverge. Therefore, for a physical quantity like energy dissipation to be well-defined and invariant under reordering of events, the underlying series must be absolutely convergent. Absolute convergence guarantees that the sum remains constant regardless of term permutation, ensuring the mathematical model accurately reflects physical reality where the total effect shouldn't depend on arbitrary sequencing.

📖 Explanation: To analyze this series, one must first test for absolute convergence by examining $\sum \frac{\ln(k)}{k^2}$. Although $\ln(k)$ grows without bound, it grows much slower than any positive power of $k$. By comparing to the convergent p-series $\sum \frac{1}{k^{1.5}}$, we see that $\frac{\ln(k)}{k^2} < \frac{1}{k^{1.5}}$ for sufficiently large $k$. Since the series of absolute values converges via direct comparison, the original alternating series converges absolutely. This avoids the unnecessary step of checking conditional convergence criteria.

📖 Explanation: This question integrates graphical interpretation with analytical definitions. The oscillating partial sums with decreasing amplitude suggest convergence of the original series (likely satisfying the Alternating Series Test). However, the infinite area under $|a_k|$ indicates that $\int |f(x)| dx$ diverges, implying $\sum |a_k|$ diverges by the Integral Test logic. Combining these two observations—convergence of the signed series and divergence of the absolute series—leads definitively to the conclusion of conditional convergence. Students must synthesize visual data with theoretical tests.

📖 Explanation: The original series is the alternating harmonic series, which converges conditionally because $\sum \frac{1}{k}$ diverges. To achieve absolute convergence, the series of absolute values must converge. Replacing $k$ with $k^2$ yields $\sum \frac{1}{k^2}$, a convergent p-series with $p=2 > 1$. Multiplying by $(-1)^k$ removes alternation but leaves the harmonic series. Removing signs yields the divergent harmonic series. Adding a constant causes divergence by the nth-term test. Only increasing the decay rate of the denominator ensures absolute summability.

📖 Explanation: This is a classic error analysis question targeting misuse of the Ratio Test. The limit $\rho = 1$ is the indeterminate case for the Ratio Test; it provides zero information about convergence or divergence. Series with $\rho=1$ can converge absolutely (e.g., $\sum 1/k^2$), converge conditionally (e.g., $\sum (-1)^k/k$), or diverge (e.g., $\sum 1/k$). The student's conclusion is unjustified because they treated an inconclusive result as a definitive classification. Further testing with other methods is strictly required.

📖 Explanation: This problem tests mixed concepts involving algebraic properties of series. Let $c_k = a_k + b_k$. If $\sum c_k$ were absolutely convergent, then $\sum |c_k|$ would converge. Since $b_k = c_k - a_k$, and both $\sum c_k$ and $\sum a_k$ would be absolutely convergent, their difference $\sum b_k$ would also have to be absolutely convergent. But this contradicts the premise that $\sum b_k$ is only conditionally convergent. Therefore, $\sum (a_k + b_k)$ cannot be absolutely convergent; it must retain the conditional nature of $b_k$.

📖 Explanation: This question links absolute convergence to the geometry of power series intervals. A fundamental theorem states that if a power series converges absolutely at a point $b$, it must converge absolutely for every point strictly closer to the center $a$ than $b$ is. This defines the interior of the interval of convergence. Conditional convergence can only potentially occur at the boundary endpoints, never in the interior if absolute convergence holds elsewhere inside. Option C is incorrect because divergence is only guaranteed outside the radius, not necessarily at every point beyond $b$.

📖 Explanation: This challenging question probes the relationship between square-summability and absolute summability. Consider $a_k = \frac{(-1)^k}{k}$. Then $a_k^2 = \frac{1}{k^2}$, which converges. However, $\sum |a_k| = \sum \frac{1}{k}$ diverges. Thus, convergence of squares does not imply absolute convergence. The statement is false. Note that the converse is true: if $\sum |a_k|$ converges, then $\sum a_k^2$ converges by comparison since $|a_k| \to 0$ implies $a_k^2 \leq |a_k|$ eventually. This asymmetry is a subtle but critical concept in analysis.

📖 Explanation: This application question contrasts practical efficiency rooted in convergence types. The alternating harmonic series converges conditionally, meaning terms decay as $1/k$, requiring millions of terms for high precision. The series for $\ln(\frac{1+x}{1-x}) = 2\sum \frac{x^{2k+1}}{2k+1}$ at $x=1/3$ involves powers of $1/9$, leading to exponential decay of terms. This rapid geometric-like decay stems from evaluating within the open interval of absolute convergence, far from the boundary singularity. Understanding this link between absolute convergence regions and computational speed is vital for numerical analysis.

📖 Explanation: This error analysis clarifies the precise output of convergence tests. The Root Test is inherently a test for absolute convergence because it evaluates $\lim |a_k|^{1/k}$. When this limit $\rho < 1$, the theorem explicitly concludes that $\sum |a_k|$ converges. Regular convergence is merely a corollary of absolute convergence. The student's hesitation reveals a misunderstanding of the test's mechanism; there is no separate 'regular convergence' version of the Root Test distinct from the absolute version. Any $\rho < 1$ guarantees absolute summability.

📖 Explanation: This conceptual question highlights the fragility of conditionally convergent series. By the Riemann Rearrangement Theorem, rearranging terms can change the sum or cause divergence, eliminating option A. Multiplying by $(-1)^k$ alters signs unpredictably, and taking absolute values yields a divergent series. However, grouping consecutive terms (associativity) preserves the sum for any convergent series, whether absolute or conditional, because the subsequence of partial sums corresponding to the groups is a subsequence of the original convergent sequence. This distinguishes safe associative operations from unsafe commutative ones in conditional contexts.

📖 Explanation: This direct recall/application hybrid tests knowledge of standard counterexamples. The conjecture fails precisely for conditionally convergent series. Option A converges absolutely. Option B converges absolutely. Option D diverges entirely. Option C, the alternating harmonic series, has terms $1/k$ that decrease monotonically to zero and the series converges by AST, but $\sum 1/k$ diverges. Thus, it satisfies all premises (convergent, terms decrease to zero) yet fails the conclusion (absolute convergence). Recognizing this canonical example is foundational for understanding the boundaries of absolute convergence theory.

📖 Explanation: This advanced application connects absolute convergence to function regularity. If Fourier coefficients are absolutely summable, the Weierstrass M-test ensures uniform convergence of the trigonometric series. Uniform limits of continuous functions (trigonometric polynomials) are continuous. Thus, absolute convergence of coefficients is a sufficient condition for continuity of the represented function. Mere square-summability ($L^2$) only guarantees mean-square convergence, allowing discontinuities. This illustrates how stronger convergence modes (absolute vs. conditional/mean-square) translate to stronger analytical properties in applied mathematics and signal processing.

📖 Explanation: This error analysis addresses asymptotic hierarchy misconceptions. While $k!$ grows fast, $k^k$ grows vastly faster. Applying the Ratio Test: $|\frac{a_{k+1}}{a_k}| = \frac{(k+1)!}{(k+1)^{k+1}} \cdot \frac{k^k}{k!} = \frac{k+1}{(k+1)^{k+1}} k^k = (\frac{k}{k+1})^k \to \frac{1}{e} < 1$. Since $1/e \approx 0.368 < 1$, the series converges absolutely. The student's intuition about factorial dominance fails when the denominator is $k^k$ rather than $c^k$. Correct asymptotic comparison is essential for accurate absolute convergence determination.

📖 Explanation: Option C presents one absolutely convergent series ($\sum (-1)^k/k^2$, since $\sum 1/k^2$ converges) and one conditionally convergent series ($\sum (-1)^k/k$, since $\sum 1/k$ diverges). This pair explicitly shows two convergent series with different convergence strengths. Option A shows two absolutely convergent series. Option B shows one conditional and one divergent. Option D shows two divergent series. Only C isolates the distinction between absolute and conditional convergence within the set of convergent series, illustrating the strict inclusion relationship.

📖 Explanation: This multi-step reasoning problem tests preservation of absolute convergence under transformations. If $\sum |a_k|$ converges, then $|a_k| \to 0$, so eventually $|a_k| < 1$. For B: $a_k^2 = |a_k|^2 \leq |a_k|$ for large $k$, so $\sum a_k^2$ converges absolutely by Direct Comparison. For A: $|a_k/k| \leq |a_k|$ for $k \geq 1$, so $\sum a_k/k$ converges absolutely by Direct Comparison. For C: $\sqrt{|a_k|} \geq |a_k|$ when $|a_k| < 1$, so comparison fails; indeed $a_k = (-1)^k/k^2$ gives $\sqrt{|a_k|} = 1/k$ which diverges. Thus both A and B are guaranteed.

📖 Explanation: This series is the Maclaurin expansion of $\cos(x)$. Applying the Ratio Test for absolute convergence: $\lim |\frac{x^{2(n+1)}}{(2(n+1))!} \cdot \frac{(2n)!}{x^{2n}}| = \lim \frac{x^2}{(2n+2)(2n+1)} = 0$ for any fixed $x$. Since the limit is 0 < 1 for all real $x$, the series converges absolutely on $(-\infty, \infty)$. Physically, cosine represents bounded oscillation for all real inputs. Students must connect the analytical result of infinite radius of absolute convergence to the physical domain of the modeled phenomenon, rejecting artificial restrictions like $|x|<1$.

📖 Explanation: The Alternating Series Test specifically requires terms to alternate in sign in a strict pattern (e.g., $(-1)^k b_k$) and have monotonically decreasing magnitude. The series $\sum \sin(k)/k$ has irregular sign changes because $\sin(k)$ oscillates non-periodically with respect to integer $k$. More importantly, AST tests for conditional convergence, never absolute convergence. To test absolute convergence, one would examine $\sum |\sin(k)|/k$, which actually diverges by comparison to $\sum 1/k$ since $|\sin(k)|$ doesn't average to zero. This highlights the limitations of AST and the complexity of non-alternating sign-varying series.

📖 Explanation: Since $|\sin(k)| \leq 1$ for all $k$, we have $|a_k \sin(k)| \leq |a_k|$. If $\sum |a_k|$ converges (absolute convergence of original series), then by the Direct Comparison Test, $\sum |a_k \sin(k)|$ must also converge. This holds regardless of the sign of $a_k$ or the irregularity of $\sin(k)$. The boundedness of sine acts as a damping factor that preserves absolute summability. This demonstrates how multiplying an absolutely convergent series by any bounded sequence maintains absolute convergence, a powerful tool in analysis.

📖 Explanation: Divergence of $\sum |a_k|$ only establishes that the series is not absolutely convergent. It does not establish that $\sum a_k$ diverges; the series could still converge conditionally. The omitted critical step is applying appropriate tests (AST, Dirichlet, Abel, etc.) to the original signed series. Many students conflate 'not absolutely convergent' with 'divergent,' but conditional convergence occupies the logical space between these states. Proper procedure demands investigating the signed series independently after absolute divergence is established.

📖 Explanation: To test absolute convergence, we examine $\sum |a_k| = \sum \frac{1}{k \ln k}$. The Integral Test applies to this positive series using $f(x) = \frac{1}{x \ln x}$. Option B is invalid because $(-1)^x$ isn't defined for real integration in this context. Option C corresponds to a different convergent series. Option D corresponds to a divergent series but with wrong integrand. Evaluating $\int \frac{dx}{x \ln x} = \ln(\ln x) |_2^\infty = \infty$, confirming $\sum |a_k|$ diverges. Thus the original series is only conditionally convergent. Correct integral setup is essential.

📖 Explanation: At the boundary of the interval of convergence, behavior at each endpoint is independent. Conditional convergence at $x=3$ tells us nothing definitive about $x=-3$. Examples exist for all three possibilities: $\sum (-1)^k x^k/(k \cdot 3^k)$ converges conditionally at both; $\sum x^k/(k \cdot 3^k)$ converges conditionally at $x=-3$ but diverges at $x=3$; modified series can yield absolute convergence at one endpoint. Students must understand that boundary analysis requires separate testing for each endpoint; symmetry is not guaranteed unless coefficients have special properties.

📖 Explanation: Decompose: $\sum a_k = \sum \frac{(-1)^k}{\sqrt{k}} + \sum \frac{1}{k^2}$. The second part converges absolutely. The first part converges conditionally (AST applies, but $\sum 1/\sqrt{k}$ diverges). The sum of a conditionally convergent series and an absolutely convergent series is always conditionally convergent. For absolute convergence, $\sum |a_k|$ would need to converge, but $|a_k| \approx 1/\sqrt{k}$ for large $k$, and $\sum 1/\sqrt{k}$ diverges. This decomposition technique is essential for analyzing composite series where direct testing is difficult.

📖 Explanation: Logarithmic growth of partial sums $S_n \sim \ln n$ is characteristic of the harmonic series $\sum 1/k$. This indicates $|a_k| \sim C/k$ asymptotically. If terms decayed as $1/k^2$, partial sums would converge to a finite value. Exponential decay would show rapid saturation. Non-zero limit would imply linear growth. The graph thus reveals that absolute values behave harmonically (divergent), while the signed series converges (likely alternating with $1/k$ envelope). Interpreting growth rates visually connects discrete series behavior to continuous asymptotic models.

📖 Explanation: The Ratio Test with $\rho < 1$ is fundamentally a test for absolute convergence; it compares $|a_k|$ to a convergent geometric series. The Alternating Series Test only requires decreasing magnitude and zero limit, establishing convergence of the signed series without addressing $\sum |a_k|$. Many conditionally convergent series satisfy AST but fail Ratio Test for absolute convergence (e.g., alternating harmonic has $\rho=1$). Understanding this distinction prevents misclassification: Ratio < 1 ⇒ Absolute; AST ⇒ Convergent (possibly conditional). This hierarchy is central to series taxonomy.

📖 Explanation: Floating-point arithmetic introduces small perturbations equivalent to reordering and modifying terms. Conditionally convergent series are unstable under such perturbations due to the Riemann Rearrangement Theorem; tiny errors can accumulate to produce wildly incorrect sums. Absolutely convergent series are robust: small perturbations yield small changes in the sum. This stability is critical in scientific computing. Additionally, absolute convergence often permits error bounding via geometric majorants, enabling reliable stopping criteria. Conditional series lack such robust error control, making them numerically hazardous despite theoretical convergence.

📖 Explanation: For absolute convergence, examine $\sum \frac{\arctan(k)}{k}$. Since $\lim_{k \to \infty} \arctan(k) = \pi/2$, we have $\frac{\arctan(k)}{k} \sim \frac{\pi}{2k}$. By Limit Comparison with $\sum 1/k$, the series of absolute values diverges. Thus, it does NOT converge absolutely. Note: The original signed series DOES converge conditionally by AST since $\arctan(k)/k$ eventually decreases to 0. Option A incorrectly assumes boundedness implies summability. Option C confuses conditional with absolute. Option D cites increase but misses the asymptotic equivalence to harmonic. Asymptotic analysis is key.

📖 Explanation: This applies known convergence benchmarks. The series $\sum \frac{1}{k (\ln k)^p}$ converges if and only if $p > 1$ (by Integral Test). Here $p=2 > 1$, so $\sum \frac{1}{k (\ln k)^2}$ converges. Since $|a_k|$ is bounded above by terms of a convergent positive series, $\sum |a_k|$ converges by Direct Comparison. Therefore, $\sum a_k$ converges absolutely. Recognizing logarithmic p-series benchmarks extends absolute convergence testing beyond standard p-series, covering important borderline cases in analysis.

📖 Explanation: Expanding absolute terms: $|a_k| = \frac{1}{k} + \frac{1}{k^2}$. Summing gives $\sum \frac{1}{k}$ (divergent) plus $\sum \frac{1}{k^2}$ (convergent). Divergent + Convergent = Divergent. So $\sum |a_k|$ diverges. Original series: $\sum \frac{(-1)^k}{k} + \sum \frac{(-1)^k}{k^2}$. First is conditional, second is absolute. Their sum converges (conditional + absolute = conditional). Both explanations A and C capture aspects of this: C provides rigorous decomposition, A provides intuitive comparison. Together they fully explain the conditional nature. Multi-faceted reasoning strengthens understanding.

📖 Explanation: Options A, B, D preserve absolute convergence: A multiplies by decaying exponential; B multiplies by bounded function; D adds absolutely convergent series. Option C: if $a_k = (-1)^k/k^2$, then $\sqrt{|a_k|} = 1/k$, and $\sum 1/k$ diverges. Taking square roots amplifies small terms, potentially destroying summability. Absolute convergence is preserved under multiplication by bounded sequences and addition of absolutely convergent series, but NOT under nonlinear operations like roots that increase term magnitude relative to original. This tests deep understanding of operational stability in absolute convergence.

📖 Explanation: Limit Comparison Test with $\rho = 0$ only allows conclusions if the comparison series CONVERGES. Here $\sum 1/k$ DIVERGES. If $|a_k|/ (1/k) \to 0$, it means $|a_k|$ is asymptotically smaller than $1/k$, but could still be $1/(k \ln k)$ which diverges, or $1/k^2$ which converges. No conclusion is possible. The student mistakenly applied the 'smaller than convergent' logic to a divergent comparator. Proper use requires matching comparator convergence/divergence status to the limit case. This is a frequent pitfall in absolute convergence testing.

📖 Explanation: Option C is geometric with ratio $1/2$, exhibiting exponential decay $(1/2)^k$, guaranteeing absolute convergence. Option A has polynomial decay $1/k^3$. Option B has super-exponential decay but is less canonical. Option D has polynomial-logarithmic decay. While B also converges absolutely, C is the prototypical example of absolute convergence driven purely by exponential base < 1, contrasting with p-series polynomial mechanisms. Recognizing decay type helps select appropriate tests: Ratio/Root for exponential, Comparison/Integral for polynomial. This classification aids efficient problem-solving strategy.

📖 Explanation: The proof uses the Squeeze Theorem or Cauchy Criterion, both relying on $-|a_k| \leq a_k \leq |a_k|$. This bounds the signed term between absolute values. If $\sum |a_k|$ converges, then partial sums of $|a_k|$ form a Cauchy sequence, and the inequality transfers this property to $\sum a_k$. Option A is false for negative terms. Option C requires $|a_k| < 1$. Option D is reverse triangle inequality. Mastery of this basic inequality underpins all theoretical connections between absolute and regular convergence, making it foundational despite simplicity.

📖 Explanation: Partial sum growth rate $S_n \sim n^\alpha$ corresponds to term decay $|a_k| \sim k^{\alpha-1}$. Here $\alpha = 1/2$, so $|a_k| \sim k^{-1/2} = 1/\sqrt{k}$. This confirms divergence of $\sum |a_k|$ (since $p=1/2 < 1$) while $\sum a_k$ converges (presumably by alternation). Connecting cumulative growth to individual term asymptotics is a powerful inverse reasoning skill. Option B would give $\ln n$ growth. Option C would give convergent partial sums. Exponential decay gives bounded partial sums. Only A matches $\sqrt{n}$ accumulation.

📖 Explanation: Term-by-term integration over unbounded domains requires uniform convergence or dominated convergence, typically ensured by absolute integrability. $\sum e^{-kx}$ on [0,∞) involves improper integrals; absolute convergence justifies swapping sum and integral via Fubini/Tonelli. Option A is bounded domain with conditional convergence (problematic but different). Option C is trivial. Option D is bounded domain within radius of convergence. Unbounded domains amplify risks of conditional convergence failures; absolute convergence provides the necessary domination condition for valid interchange. This connects series theory to measure-theoretic integration principles in advanced applications.