📝 Maclaurin series Taylor series examples (35 MCQs)

📖 Explanation: While the derivative method is theoretically valid, computing higher derivatives of composite functions like $\ln(1+x^2)$ is prone to algebraic error and computationally expensive. Substituting $u=x^2$ into the known expansion for $\ln(1+u)$ is a higher-order strategy that leverages existing knowledge. This question tests the conceptual understanding of when to apply direct definition versus algebraic manipulation of known series, emphasizing efficiency and error reduction in mathematical modeling.

📖 Explanation: Differentiation of a power series preserves the radius of convergence $R$, but endpoint behavior can change. The original series converges conditionally at $x=2$ (alternating harmonic) and diverges at $x=4$ (harmonic). The differentiated series $\sum (x-3)^{k-1}$ is geometric with ratio $|x-3|$; it diverges at both endpoints $x=2$ and $x=4$. Thus, the open interval $(2,4)$ remains, but conditional convergence at the boundary is lost. This tests deep understanding of endpoint analysis versus radius preservation.

📖 Explanation: Expanding $(1-u)^{-1/2}$ where $u=v^2/c^2$ gives $1 + \frac{1}{2}u + \frac{3}{8}u^2 + \dots$. Subtracting 1 and multiplying by $m_0 c^2$ yields $\frac{1}{2}m_0 v^2 + \frac{3}{8}m_0 v^4/c^2 + \dots$. The first term is Newtonian KE. The second term is the leading-order relativistic correction. This requires applying binomial series to a physics model and interpreting the physical significance of specific terms in an asymptotic expansion, bridging pure math and applied science.

📖 Explanation: While the integrated series $\sum (-1)^k / [(2k+1)k!]$ is indeed alternating, students often blindly apply the error bound without verifying the decreasing condition $|a_{n+1}| \leq |a_n|$. In this case it holds, but the *reasoning* is incomplete without verification. More subtly, if the upper limit were variable or large, monotonicity might fail initially. This question targets the misconception that 'alternating' automatically implies 'estimable by next term' without checking all hypotheses, promoting rigorous error analysis over rote procedure.

📖 Explanation: At an inflection point, f''(a)=0. A quadratic approximation P_2(x) = f(a)+f'(a)(x-a)+0 reduces to the linear tangent, matching $P_1$. However, if a student incorrectly computes $P_2$ assuming non-zero curvature or misinterprets the graph, discrepancies arise. But theoretically, $P_1$ and $P_2$ coincide. The key insight is recognizing that at inflection points, even-degree Taylor polynomials may degenerate. This tests graphical interpretation of derivative conditions and understanding how polynomial degree relates to local geometry, specifically identifying when higher-order terms vanish.

📖 Explanation: First expand $\sin(u) = u - u^3/6 + \dots$ with $u=t^2$: $\sin(t^2) = t^2 - t^6/6 + \dots$. Integrating term-by-term: $f(x) = x^3/3 - x^7/(6 \cdot 7) + \dots = x^3/3 - x^7/42 + \dots$. The coefficient is $-1/42$. This combines substitution, series expansion, and integration. It avoids brute-force differentiation of an integral function, testing strategic problem-solving and synthesis of multiple calculus operations within the power series framework.

📖 Explanation: This classic counterexample shows that $C^\infty$ does not imply analyticity. The Taylor series is identically zero (since all $f^{(n)}(0)=0$), so it converges everywhere to the zero function. But $f(x) > 0$ for $x \neq 0$, so the series equals $f$ only at the center. This challenges the assumption that infinite differentiability guarantees power series representation, highlighting the distinction between smooth and analytic functions—a profound conceptual point often missed in computational courses.

📖 Explanation: While formal division yields correct coefficients, the resulting series for $\tan(x)$ has finite radius $\pi/2$ due to poles where $\cos(x)=0$, unlike $\sin$ and $\cos$ which converge everywhere. Students might assume operations on entire functions yield entire functions. This tests understanding that algebraic operations on series preserve validity only where denominators are non-zero and within convergence domains, linking complex singularities to real series behavior.

📖 Explanation: The binomial expansion $(1-u)^{-1/2} = \sum c_n u^n$ has positive increasing coefficients $c_n$. When $u=k^2 \sin^2 \phi \ll 1$, terms decay fast. When $k \approx 0.7$, $u$ approaches 0.5, and combined with growing $c_n$, convergence slows dramatically. This requires analyzing rate of convergence dependent on parameter magnitude, connecting series theory to practical numerical modeling trade-offs in physics.

📖 Explanation: $\sqrt{x}$ expanded about $x=1$ has radius 1, so interval is $(0,2)$. At $x=0$, the series becomes alternating with terms $\sim 1/\sqrt{n}$, which converges conditionally. But $\sqrt{x}$ is defined at 0. Wait—actually the issue is that $\sqrt{x}$ isn't analytic at 0, but the series about 1 *does* converge at 0. The trick: option C says $0<x<2$, excluding 0 unnecessarily? No—the real catch is that some series match derivatives but fail at endpoints. Actually, all listed are standard. Re-evaluating: perhaps the distractor is subtle. Correct answer hinges on knowing exact intervals. $\sqrt{x}$ series about 1 converges at x=0 to 0, so (0,2) is too restrictive; [0,2) is correct. Thus stating (0,2) misses valid endpoint. Tests precise interval knowledge.

📖 Explanation: For $x=0.1$, the Maclaurin series for cosine is alternating with decreasing terms. The Alternating Series Estimation Theorem states error magnitude is less than first omitted term. Student's reasoning is correct and complete for this specific value. Distractors target overgeneralization fears. This confirms proper application context: alternating property holds locally for small x, and monotonic decrease is obvious here. Reinforces when simple bounds suffice versus when Lagrange remainder is mandatory.

📖 Explanation: Identical low-degree Taylor polynomials imply matching derivatives at 0, ensuring local similarity. Global divergence arises because remainders $R_n(x)$ grow differently as $|x|$ increases, especially outside convergence radii or near singularities. This illustrates the fundamental limitation of local approximations and the importance of remainder analysis for extrapolation. Graphs visualize this transition from agreement to divergence, reinforcing that series equality at a point doesn't imply functional equality globally.

📖 Explanation: Series substitution: $(1+x+x^2/2+\dots -1 -x)/x^2 = 1/2 + x/6 + \dots \to 1/2$. This reveals not just the limit but the asymptotic behavior $\approx 1/2 + x/6$, showing linear correction. L'Hôpital gives only the limit. This compares methods qualitatively, highlighting series' advantage in extracting higher-order information beyond mere limit values, crucial for sensitivity analysis in applications.

📖 Explanation: Convergence at $x=b$ implies radius $R \geq b-a$. The symmetric point $x=a-(b-a)$ is at distance $b-a$ left of center. Within open interval $(a-R, a+R)$, convergence is absolute. At boundary points distance R away, convergence may be conditional or absolute. Since $b$ is a boundary candidate, the mirror point is also boundary. We know it converges (by symmetry of radius), but not necessarily absolutely. Tests precise understanding of interval structure versus absolute/conditional distinction at endpoints.

📖 Explanation: Power series about 0 converges only for $|s|<1$. High frequency means large $|s|$, where series diverges. Stability analysis requires behavior near imaginary axis including large values. This links convergence radius to physical domain validity. Students must recognize that series expansions are local tools; global system properties require other representations (partial fractions, pole-zero plots). Tests transfer of abstract convergence concepts to engineering contexts.

📖 Explanation: $S(x)S(y) = \sum \sum x^k y^m /(k!m!) = \sum_n (x+y)^n/n!$ via binomial theorem on coefficients. For $C(x)$, product doesn't simplify to $C(x+y)$ because coefficients lack combinatorial identity matching addition formula. This probes deep algebraic structure underlying special functions: exponential generating functions encode additive structure via factorial denominators. Challenges students to derive functional equations from series definitions rather than memorizing identities.

📖 Explanation: Alternating series bound assumes exact terms. In practice, finite precision causes round-off. With 5 terms involving factorials and powers, cancellation and rounding accumulate. Also, if implementation truncated intermediate results prematurely, error compounds. This integrates numerical analysis with series theory, warning that theoretical bounds assume ideal computation. Real-world application requires considering machine epsilon alongside truncation error.

📖 Explanation: $\arctan(x)$ series: at $x=1$, alternating harmonic-like $\sum (-1)^n/(2n+1)$ converges conditionally. At $x=-1$, similarly. Option A ($\ln(1+x)$) converges conditionally at $x=1$ but diverges at $x=-1$. Option B converges absolutely at both. Option D diverges at $x=-1$ (not alternating). Tests precise recall of endpoint behaviors for standard series, foundational for selecting appropriate expansions in boundary-value problems.

📖 Explanation: On real line, $1/(1+x^2)$ is $C^\infty$. But as complex function, poles at $z=\pm i$ distance 1 from origin. Radius of convergence equals distance to nearest singularity in complex plane. This connects real series behavior to complex analysis, explaining seemingly arbitrary finite radii for benign-looking real functions. Crucial for understanding limitations of real-variable intuition and necessity of complex perspective in series theory.

📖 Explanation: $85^\circ$ is near $\pi/2$. Maclaurin at 0 needs many terms for accuracy at 1.48 rad. Taylor at $\pi/2$ uses small $(x-\pi/2)$. Alternatively, $\sin(85^\circ)=\cos(5^\circ)$, and $5^\circ \approx 0.087$ rad is tiny, so Maclaurin for cosine converges extremely fast. Both exploit proximity to expansion point. Tests strategic selection of expansion center based on argument size, optimizing computational effort.

📖 Explanation: $g(x) = \sum a_n (x^2)^n = \sum a_n x^{2n}$. Converges when $|x^2| < 2 \Rightarrow |x| < \sqrt{2}$. Substitution $u=x^2$ maps disk of radius $\sqrt{R}$ to disk of radius R in u-plane. Tests understanding how variable transformations affect convergence domains, requiring algebraic manipulation of inequality rather than mechanical formula application.

📖 Explanation: Sharp divergence beyond |x|=1 suggests either singularity limiting radius or explosive growth of higher derivatives making remainder large even within radius. Graph alone can't distinguish, but indicates breakdown of low-order approximation. Tests interpreting visual discrepancy as diagnostic tool for function properties, linking graphical observation to analytical concepts of remainder and radius.

📖 Explanation: ASE theorem gives $|error| \leq |a_{n+1}|$, which is worst-case. Actual error is $|\sum_{k=n+1}^\infty (-1)^{k+1} x^k/k|$, typically smaller due to cancellation in tail. Bound is conservative. This clarifies that error bounds guarantee safety, not precision. Important for engineering where overestimation wastes resources. Tests nuanced understanding of inequality versus equality in error estimation.

📖 Explanation: $\limsup |a_n|^{1/n} = \max(\limsup_{even} 2, \limsup_{odd} 3) = 3$. So $R=1/3$. Root test uses limsup, not limit, precisely for oscillating coefficients. Even subsequence gives 2, odd gives 3; supremum dominates. Tests mastery of Cauchy-Hadamard theorem with non-monotonic coefficients, challenging assumption that coefficient patterns are regular.

📖 Explanation: Example: $\sum x^n$ diverges at x=1. Integrated: $\sum x^{n+1}/(n+1)$, which is $-\ln(1-x)$, converges conditionally at x=1. Factor $1/(n+1)$ turns divergent harmonic-like series into convergent alternating/harmonic variant. Integration smooths singularities, potentially extending convergence to boundary. Contrasts with differentiation, which amplifies terms. Tests understanding of how calculus operations alter series convergence characteristics at critical points.

📖 Explanation: Let $x=E/kT$. High T ⇒ small x ⇒ Maclaurin $e^{-x} \approx 1-x+\dots$ valid. Low T ⇒ large x ⇒ series diverges. Physical regime dictates mathematical tool validity. Tests recognizing domain restrictions of series approximations in scientific models, preventing misuse outside convergence region.

📖 Explanation: $\sin(x) = \sum (-1)^n x^{2n+1}/(2n+1)!$, odd powers, infinite radius. $\ln(1+x)$ has all powers, R=1. $\tan(x)$ odd but finite R. $\sqrt{1+x}$ all powers, R=1. Basic identification reinforced through elimination, ensuring foundational knowledge supports higher-order tasks.

📖 Explanation: Start with $\sum x^n = 1/(1-x)$. Differentiate: $\sum n x^{n-1} = 1/(1-x)^2$. Multiply by x: $\sum n x^n = x/(1-x)^2$. Set x=1/2: $(1/2)/(1/2)^2 = 2$. Efficiently leverages calculus on generating functions. Tests strategic manipulation of series to evaluate numeric sums, connecting discrete summation to continuous function operations.

📖 Explanation: $e^{2rt} = 1 + 2rt + (2rt)^2/2 + \dots \neq 2(1+rt)$. Nonlinearity means superposition fails. Linear model valid only locally; parameter scaling invalidates approximation. Tests understanding of linearization limits in dynamic systems, crucial for sensitivity analysis where parameters vary.

📖 Explanation: Integrated series: terms $a_n = 1/[(2n+1)(2n+1)!]$. Clearly positive, decreasing, →0. Alternating sign from $(-1)^n$. ASE applies. Student correct. Distractors target common doubts about post-integration series properties. Reinforces that integration preserves alternating/decreasing structure for well-behaved integrands, validating error estimation.

📖 Explanation: For $|x|>1$, terms grow, partial sums oscillate with increasing amplitude, never approaching limit. Visual signature of divergence. Contrasts with Gibbs phenomenon (overshoot at jump discontinuities in Fourier series). Tests distinguishing divergence artifacts from other oscillatory behaviors via graphical diagnosis.

📖 Explanation: Analytic functions vanishing on open set vanish identically on connected domain. So $f \equiv 0$ on $(-R,R)$, implying all $a_n=0$. Subinterval suffices because analytic continuation is unique. Tests deep property of power series: local zero implies global zero within radius. Contrasts with smooth non-analytic functions. Olympiad-level insight into rigidity of analytic functions.

📖 Explanation: Center at 1. Nearest non-differentiable point is x=0, distance 1. So R=1, interval (0,2). For x<0, series diverges even though function exists. Highlights that Taylor series depend on local smoothness, not global definition. Tests understanding that nondifferentiability acts as barrier to analytic continuation, even for continuous functions.

📖 Explanation: $e^r = 1+r+r^2/2+\dots$. $(1+r/n)^n = \exp(n \ln(1+r/n)) = \exp(r - r^2/(2n)+\dots) = e^r \cdot e^{-r^2/(2n)+\dots} \approx e^r (1 - r^2/(2n))$. Difference $\approx e^r \cdot r^2/(2n)$. Series expansion quantifies discretization error. Tests asymptotic analysis combining multiple expansions to extract leading error term, vital for numerical method validation.

📖 Explanation: $t \sin(t^3) = t^4 - t^{10}/6 + \dots$. Integral: $x^5/5 - x^{11}/66 + \dots$. No $x^7$ term ⇒ coefficient 0 ⇒ $f^{(7)}(0)/7! = 0$. Elegant extraction via series bypasses messy differentiation. Tests strategic use of series to read off derivative values from coefficients, showcasing power of series as derivative encoders.