AP Unit 10 Problems

(a) For a_n=(−1)^(n+1)/2^n, establish absolute convergence by evaluating the magnitude series. Then find the original signed sum S.

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(b) Regroup a_n=(−1)^(n+1)/2^n into consecutive finite blocks of lengths 1,2,3,…, using every term once in its original order. Compute b₁,b₂,b₃ and their partial sums. Prove that after K blocks the endpoint is m_K=K(K+1)/2 and that the grouped partial sums approach 1/3. Sketch S_N for N=1,…,10, and mark N=1,3,6,10 as the first four grouped endpoints. You may use S_N=(1/3)[1−(−1/2)^N].

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(c) For the absolutely convergent series with a_n=(−1)^(n+1)/2^n, decide whether the rearranged order a₂,a₁,a₄,a₃,a₆,a₅,…, using every term exactly once, preserves the original sum 1/3. Justify the conclusion under the absolute-convergence theorem, and verify the limit directly using even and odd rearranged partial sums. Explain how this rearrangement differs from consecutive finite regrouping.

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(a) For the conditionally convergent alternating harmonic series A, form consecutive pairs b_k=1/(2k−1)−1/(2k). Find its first two grouped partial sums and prove the regrouped series still sums to ln 2. State exactly which convergence and grouping hypotheses your proof uses.

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(b) For D=1−1+1−1+…, compute S_N for N=1,…,8 and sketch these points on partial-sum axes. Determine the limits of the even and odd subsequences. Evaluate the grouped partial sums for (1−1)+(1−1)+… and for 1+(−1+1)+(−1+1)+… . Does either grouping prove an ordinary sum for D? Explain.

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(c) An analyst argues that because the alternating harmonic series converges, both consecutive finite grouping and every rearrangement using each term once must preserve ln 2. Identify the justified portion and the unjustified portion. Then state separately what absolute convergence would guarantee about regrouping and about rearrangement.

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(a) For e^x centered at 0, derive the coefficient of x^k from the kth derivative at 0. Write T₁, T₃ and T₅, then evaluate each at x=−1,0,1. Give exact polynomial values.

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(b) Using the supplied remainder estimate, prove that T_n approaches e^x for every x in the entire interval [−1,1] as n increases. Establish a bound independent of x, show that it tends to zero, and explain why one sufficiently large degree works for any prescribed positive tolerance at every point in the interval.

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(c) Sketch e^x and T₁,T₃,T₅ on the common interval [−1,1], with labeled axes and a legend. Use their three supplied or calculated values at −1,0,1 and the exact formulas to show the curve shapes. Explain what the plot illustrates and why it cannot replace the interval proof in part b.

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(d) Find the smallest degree certified by B_n=3/(n+1)! to give absolute error at most 0.001 throughout [−1,1]. Compare the immediately preceding degree, and state whether this certificate proves that the actual error decreases at each degree for each x.

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(a) Compute the derivative values at 0 and construct T₀,T₁,T₂. Evaluate f and those polynomials at x=3/4, then compute each actual absolute error. Does increasing degree from 0 to 1 improve the approximation at this point?

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(b) For f(x)=1+x−x², determine T_n for every n≥2. Prove increasing-degree convergence to f throughout [−1,1] using an interval-wide remainder bound. Explain why the pointwise error increase in part a does not contradict this conclusion.

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(c) Draw the actual absolute error versus degree at x=3/4 for degrees 0,1,2, using a labeled linear vertical scale. Mark what happens for all degrees n≥2. Interpret the graph as evidence against the claim that each larger degree must improve every pointwise approximation.

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(a) State the general definition of a real-coefficient power series centered at an arbitrary real number c. Specify the allowed index and powers, the nature of the coefficient sequence, and what must remain independent of x. Explain how a series starting at n=1 and a finite polynomial fit the definition.

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(b) Rewrite P(x)=2−√2(x+1)²+Σ_(k=3)^∞ [((−1)^k)/(k+1)](x+1)^(2k) as Σ_(n=0)^∞ a_n(x−c)^n. Identify c; list a₀ through a₈, including every zero; and give a rule for all later coefficients. Explain why √2 and the missing powers cause no problem for the definition.

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(c) Classify B(x)=Σ_(n=0)^∞ (x+1)^(n+1/2)/2^n, C(x)=Σ_(n=1)^∞ 3^(−n)(x+1)^(−n), and D(x)=Σ_(n=0)^∞ cos(n)(x+1)^n as displayed power-series expansions at center −1, or as expressions that fail this definition. Give the specific reason in each case. Do not infer that a function can never have some different local expansion.

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(d) Evaluate P at its center c=−1 by separating its constant term. Explain why the center value is neither zero nor an undefined 0⁰ computation.

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(a) For F(x)=7+Σ_(n=1)^∞ n!(x−2)^n, first evaluate the series at x=2. Then, for any fixed x≠2, compute the ratio of consecutive tail-term magnitudes and use it to establish divergence. State the radius and complete convergence set, and explain why a ratio calculation dividing by zero must not be used at the center.

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(b) For G(x)=−3+Σ_(n=1)^∞ ((x−2)/2)^n, determine the radius and open convergence interval. Test both finite boundary points using the original terms, give the complete interval, and evaluate G at the center.

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(c) For the same centered expressions F,G,H, write H as an infinite power series by giving its coefficient sequence. Determine H’s radius and convergence set. Draw three complete convergence number lines for F,G,H on a common scale, labeling center 2, all finite endpoints and their inclusion or exclusion. Use these examples to explain the center-only, finite-interval and all-real-number alternatives.

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(a) Given the bound |e^x−T_n(x)|≤3/(n+1)! throughout [−1,1], which conclusion necessarily follows as the degree increases?

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(a) For F(x)=7+Σ_(n=1)^∞ n!(x−2)^n, which radius, convergence set and center value are correct?

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(a) Determine the center, radius, and interval of convergence of F. Show the ratio limit and both endpoint tests rather than treating a ratio-test limit of 1 as a decision.

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(b) Integrate term by term from −1 to x to construct the complete infinite series for G, including its constant value at the center. Determine its radius and explicitly compare it with F’s radius. Give either a new ratio-limit calculation or the radius-preservation theorem with its interior validity condition.

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(c) Test both endpoints of G’s series and give its full interval. On a common number line, draw the complete F and G intervals with labeled center, radius, and open or filled endpoints. Explain why adding x=2 to G’s interval does not give a radius larger than F’s.

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(d) Differentiate F term by term to find its derivative series, then test its endpoints. Compare the radii of F, F′, and G, explaining what differentiation and integration preserve and what each operation requires you to retest. Extend your number-line comparison to include F′.

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(a) Choose the correct radius and endpoint conclusion for H, and explain the relevant endpoint tests.

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(a) Write explicitly the Maclaurin identity for 1/(1−u), identify its first term and common ratio, and derive it from the finite sum through uᴺ. State its representation domain and test u=−1 and u=1.

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(b) Use that identity to construct the complete infinite series for f, state its radius and interval, and test both endpoints. Evaluate the statement “f(2) exists, so its Maclaurin series must converge at 2.”

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(c) At x=1/2, use the first three nonzero terms to approximate f. Give a guaranteed alternating error bound and compare it with the exact error using f(1/2)=4/13. Explain why the finite polynomial is not the infinite series.

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(a) Write the sine foundation explicitly and use it to construct a complete indexed infinite Maclaurin series for S, not merely a finite polynomial. Give its first four nonzero terms, radius, representation interval, and explanation of the value at zero.

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(b) Use the first three nonzero terms to estimate S(1/6). Verify the alternating magnitude condition, give a guaranteed error bound, and compare the bound with the observed difference from the supplied check. State whether the partial sum is above or below S(1/6).

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(c) Explain why multiplying the original sine-series coefficients by only 3 gives the wrong construction for sin(3x), and why dividing sin(3x) by x does not force a negative-power term here. Use the general term and the zero-value extension in your explanation.

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(a) Write the exponential foundation explicitly, substitute −2t², and integrate term by term to construct the complete infinite Maclaurin series for W. Give the first four nonzero terms, fix the constant using W(0), and justify its all-real representation interval.

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(b) Estimate W(1/2) using the first three nonzero terms. Justify an alternating error bound, compare it with the supplied independent integral, and state whether the partial sum is an upper or lower estimate.

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(c) Differentiate your infinite series to check that it recovers e^(−2x²), and explain why W is increasing and odd. Use the supplied integrand graph to interpret W(1/2) as an area, and distinguish that area from the polynomial approximation in part b.

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