451. Which of the following statements concerning the sequence \( \{a_n\} = \frac{n}{2n^2-3} \) is true?
451. Which of the following statements concerning the sequence \( \{a_n\} = \frac{n}{2n^2-3} \) is true?
答案： ( B )
\( a_n = \frac{n}{2n^2-3} \to 0 \) as \( n \to \infty \). For \( n \geq 2 \), \( a_n > \frac{1}{2n} \), so \( \sum_{n=2}^\infty a_n > \frac{1}{2}\sum_{n=2}^\infty \frac{1}{n} = \infty \); series diverges.
452. If the \( n \)th partial sum of \( \sum_{i=1}^\infty a_i \) is \( s_n = \frac{3n+1}{2n-5} \), find \( a_5 \).
答案： ( C )
\( a_5 = s_5 - s_4 = \frac{16}{5} - \frac{13}{3} = \frac{48-65}{15} = -\frac{17}{15} \).
453. Leibniz's Theorem says \( \sum_{n=1}^\infty (-1)^{n+1}\frac{n+1}{n} \) will converge because I. \( \frac{n+1}{n} \) is positive; II. \( \frac{n+2}{n+1} \leq \frac{n+1}{n} \); III. \( \lim_{n\to\infty}\frac{n+1}{n} = 1 \).
答案： ( D )
For alternating series to converge, the limit in III must be 0, not 1. So the series diverges.
454. Find the sum of \( 4 + \frac{4}{3} + \frac{4}{9} + \frac{4}{27} + \cdots \).
答案： ( D )
Geometric: \( a = 4 \), \( r = \frac{1}{3} \); sum \( = \frac{4}{1-1/3} = \frac{4}{2/3} = 6 \).
455. Express \( 1.3\overline{12} \) as a ratio of two integers.
答案： ( D )
\( 1.3\overline{12} = \frac{13}{10} + \frac{12}{990} = \frac{13\cdot 99+12}{990} = \frac{433}{330} \).
456. Is \( \sum_{n=1}^\infty 3^{2n}2^{3-3n} \) convergent or divergent? If convergent, what is the sum?
答案： ( D )
General term \( = 9^n \cdot 8/2^{3n} = 8(9/8)^n \); \( |r| = 9/8 > 1 \), so divergent.
457. Find the sum of \( \sum_{i=1}^\infty \left(\frac{1}{i} - \frac{1}{i+2}\right) \).
答案： ( A )
Telescoping: \( s_n = 1 + \frac{1}{2} - \frac{1}{n+1} - \frac{1}{n+2} \to 1 + \frac{1}{2} = \frac{3}{2} \) as \( n \to \infty \).
458. If \( \sum_{n=1}^\infty a_n = 3 \) and \( \sum_{n=1}^\infty b_n = 7 \), find \( \sum_{n=1}^\infty (6a_n - 2b_n) \).
答案： ( A )
\( \sum(6a_n - 2b_n) = 6\sum a_n - 2\sum b_n = 6(3) - 2(7) = 4 \).
459. Using Limit Comparison with \( \frac{1}{n} \): \( \sum_{n=1}^\infty \frac{3n+1}{(n+1)^2} \) will converge because I. series behaves like \( 1/n \); II. limit is 3.
答案： ( D )
Limit 3 with \( \sum \frac{1}{n} \) (divergent) implies both diverge or both converge; \( \sum 1/n \) diverges, so the series diverges.
460. Determine whether \( \sum_{n=2}^\infty \frac{5}{n^2-n} \) converges or diverges.
答案： ( A )
Limit comparison with \( b_n = \frac{5}{n^2} \): \( \frac{a_n}{b_n} = \frac{n^2}{n^2-n} \to 1 \); \( \sum \frac{5}{n^2} \) converges, so the series converges.
461. Determine whether \( \sum_{n=1}^\infty \frac{3n+7}{5n-2} \) converges or diverges.
答案： ( B )
\( \frac{3n+7}{5n-2} \to \frac{3}{5} \neq 0 \) as \( n \to \infty \); nth term test implies divergence.
462. Determine whether \( \sum_{n=1}^\infty \frac{\ln n}{n^2} \) converges or diverges.
答案： ( A )
Integral test: \( \int_1^\infty \frac{\ln x}{x^2}\,dx \) converges (e.g. by parts); so the series converges.
463. What will \( \sum_{n=0}^\infty \frac{4^n}{5^n+1} \) do?
答案： ( B )
Ratio test gives limit \( < 1 \), so the series converges.
464. Determine whether \( \sum_{n=1}^\infty \sin\left(\frac{1}{n^2}\right) \) converges or diverges.
答案： ( A )
Limit comparison with \( 1/n^2 \): \( \frac{\sin(1/n^2)}{1/n^2} \to 1 \); \( \sum 1/n^2 \) converges \( \Rightarrow \) series converges.
465. Determine whether \( \sum_{n=1}^\infty \frac{n^n}{n!} \) converges or diverges.
答案： ( B )
\( \frac{n^n}{n!} > 1 \) for \( n \geq 1 \); terms do not tend to 0 \( \Rightarrow \) diverges.
466. Determine whether \( \sum_{n=2}^\infty (-1)^n \frac{1}{\ln n} \) converges or diverges.
答案： ( A )
Alternating: \( 1/\ln n \to 0 \) and \( 1/\ln(n+1) < 1/\ln n \); by AST, converges.
467. Determine whether \( \sum_{n=1}^\infty (-1)^{n-1}\frac{e^{1/n}}{\sqrt{n}} \) converges or diverges.
答案： ( A )
Alternating: \( b_n = e^{1/n}/\sqrt{n} \to 0 \) and is decreasing for \( n \geq 1 \); AST \( \Rightarrow \) converges.
468. The series \( \sum_{n=0}^\infty \frac{|\sin x|^n}{n!} \): I. converges because \( \frac{|\sin x|^n}{n!} \leq \frac{1}{n!} \); II. converges because \( \sum_{n=0}^\infty \frac{1}{n!} = e \).
答案： ( C )
Both I and II hold; \( |\sin x|^n/n! \leq 1/n! \) and \( \sum 1/n! \) converges.
469. \( \sum_{n=0}^\infty a_n \) is defined by \( a_0 = 0 \), \( a_{n+1} = \frac{\sin n + n}{2n+1} a_n \). Determine whether it converges or diverges.
答案： ( A )
Ratio: \( \left|\frac{a_{n+1}}{a_n}\right| = \frac{\sin n+n}{2n+1} \leq \frac{n+1}{2n+1} \to \frac{1}{2} < 1 \); ratio test \( \Rightarrow \) converges.
470. Determine whether \( \sum_{n=0}^\infty n^2 e^{-2n} \) converges or diverges.
答案： ( A )
Integral test: \( \int_1^\infty x^2 e^{-2x}\,dx \) converges (e.g. by parts); series converges.
471. Determine whether \( \frac{1}{2} - \frac{2}{5} + \frac{3}{10} - \frac{4}{17} + \cdots \) is absolutely convergent, conditionally convergent, or divergent.
答案： ( B )
Series is \( \sum (-1)^{n-1}\frac{n}{n^2+1} \); AST gives convergence; absolute series diverges (compare \( 1/n \)); conditionally convergent.
472. Determine whether \( \sum (-1)^{n+1}(\frac{n}{2n+1})^n \) is absolutely convergent, conditionally convergent, or divergent.
答案： ( A )
Root test: nth root of term tends to \( 1/2 < 1 \); absolutely convergent.
473. For \( \sum (-1)^{n+1}\frac{1}{n^2} \), truncation error after fifth term is less than \( 1/36 \) because: I. terms alternate; II. terms decrease; III. terms tend to 0.
答案： ( D )
I, II, and III are all true and required for the alternating series error bound.
474. For what \( x \) is \( \sum_{n=1}^\infty \frac{x^n}{n 2^n} \) convergent?
答案： ( B )
Ratio test: \( |x|/2 < 1 \) so \( |x| < 2 \); at \( x=2 \) diverges; at \( x=-2 \) converges; hence \( [-2,2) \).
475. Radius of convergence of \( \sum_{n=0}^\infty n^3(x-4)^n \)?
答案： ( B )
Ratio: \( |x-4| \cdot (n+1)^3/n^3 \to |x-4| < 1 \); \( R = 1 \).
476. Interval of convergence of \( \sum_{n=3}^\infty \frac{x^{2n}}{(\ln n)^n} \)?
答案： ( C )
Root test: \( |x|^2/\ln n \to 0 \) for all \( x \); converges for all \( x \).
477. Power series for \( \frac{x}{2+3x} \) and interval of convergence?
答案： ( D )
\( \frac{x}{2+3x} = (x/2)\sum (-3x/2)^n \) for \( |3x/2| < 1 \); interval \( (-2/3, 2/3) \).
478. Power series for \( f(x) = \ln(1-2x) \) and radius \( r \)?
答案： ( A )
\( \ln(1-u) = -\sum u^n/n \) for \( |u|<1 \); \( u = 2x \) gives \( r = 1/2 \).
479. Truncate \( \sum (-1)^n \frac{x^{2n}}{(2n)!} \) after five terms; error is less than or equal to?
答案： ( B )
Error bounded by sixth term: \( |x|^{10}/(10!) = x^{10}/3628800 \).
480. Interval of convergence of \( \sum_{n=1}^\infty n^n x^{2n} \)?
答案： ( A )
Root test: \( n^{1/n}|x|^2 \to \infty \) for \( x \neq 0 \); converges only at \( x = 0 \).
481. Let \( s = 10 + 4 + \frac{8}{5} + \frac{16}{25} + \frac{32}{125} + \cdots \). Find the sum of the first 10 terms.
答案： ( B )
Geometric \( a = 10 \), \( r = 2/5 \); \( s_{10} = 10\frac{1-(2/5)^{10}}{1-2/5} \approx 16.665 \).
482. Find the Maclaurin series of \( f(x) = x^2 e^{2x} \) and its radius of convergence \( r \).
答案： ( C )
\( e^{2x} = \sum_{n=0}^\infty \frac{(2x)^n}{n!} \); \( x^2 e^{2x} = \sum_{n=0}^\infty \frac{2^n}{n!}x^{n+2} \); \( r = \infty \).
483. Find the degree 2 Maclaurin polynomial for \( f(x) = \ln\frac{x+1}{(x+2)^2} \).
答案： ( D )
\( f(x) = \ln(x+1) - 2\ln(x+2) \); \( f(0) = -2\ln 2 \), \( f'(0) = 0 \), \( f''(0) = -1/2 \); \( P_2 = -2\ln 2 - \frac{1}{4}x^2 \).
484. If \( f \) is approximated by the third-order Taylor \( 4 - 3(x-2) + 2(x-2)^2 - 7(x-2)^3 \) at 2, find \( f'''(2) \).
答案： ( A )
Coefficient of \( (x-2)^3 \) is \( f'''(2)/3! = -7 \) \( \Rightarrow f'''(2) = -42 \).
485. Find a Maclaurin series for \( f(x) = \cos^2 x \).
答案： ( D )
\( \cos^2 x = \frac{1+\cos 2x}{2} = \frac{1}{2} + \sum_{n=0}^\infty \frac{(-1)^n 2^{2n-1}}{(2n)!}x^{2n} \).
486. Use Maclaurin series to approximate \( \int_0^1 \frac{e^{-x}-1}{x}\,dx \) to three decimal places.
答案： ( B )
\( \frac{e^{-x}-1}{x} = -1 + \frac{x}{2} - \frac{x^2}{6} + \cdots \); integrate term-by-term; \( \approx -0.796 \).
487. Find the Taylor series for \( f(x) = 1/\sqrt{x} \) centered at \( x = 4 \).
答案： ( A )
Taylor at 4: \( f^{(n)}(4)/n! \cdot (x-4)^n \); pattern with odd double factorials and \( 2^{3n+1} \) matches (A).
488. Find the Taylor series for \( f(x) = \cos(\pi x) \) centered at \( x = 3 \).
答案： ( B )
At \( x = 3 \): \( \cos(\pi x) = \cos(3\pi + \pi(x-3)) = -\cos(\pi(x-3)) \); series in \( (x-3) \) with \( (-1)^{n+1}\pi^{2n}/(2n)! \).
489. Find the sum of \( \sum_{n=0}^\infty \frac{(-1)^n 2^n}{3^n n!} \).
答案： ( D )
\( \sum_{n=0}^\infty \frac{(-2/3)^n}{n!} = e^{-2/3} \).
490. Find the first three nonzero terms in the Maclaurin series of \( f(x) = \frac{e^x}{1-x} \).
答案： ( C )
\( e^x/(1-x) = (1+x+x^2/2+\cdots)(1+x+x^2+\cdots) = 1 + 2x + (1+1+1/2)x^2 + \cdots = 1 + 2x + \frac{5}{2}x^2 + \cdots \)
491. Approximate \( \cos(9^\circ) \) accurate to three decimal places.
答案： ( D )
\( 9^\circ = \pi/20 \) rad; use Maclaurin \( \cos x = 1 - x^2/2! + x^4/4! - \cdots \) at \( x = \pi/20 \); \( \cos(\pi/20) \approx 0.987 \).
492. Find the sum of \( \frac{\pi}{3} - \frac{(\pi/3)^3}{3!} + \frac{(\pi/3)^5}{5!} - \cdots \).
答案： ( B )
Series is \( \sin(\pi/3) = \sqrt{3}/2 \) (Maclaurin for \( \sin x \) at \( x = \pi/3 \)).
493. Evaluate \( \int \frac{\cos(x^3)-1}{x}\,dx \) as an infinite series.
答案： ( D )
\( \cos(x^3)-1 = \sum_{n=1}^\infty \frac{(-1)^n x^{6n}}{(2n)!} \); divide by \( x \), integrate: \( \sum \frac{(-1)^n}{6n(2n)!}x^{6n} + c \).
494. Use the fourth-degree Taylor polynomial \( T_4(x) \) to approximate \( \ln 2 \).
答案： ( C )
\( \ln(1-x) = -\sum x^n/n \); \( \ln 2 = \ln(1-(-1)) = f(-1) \approx T_4(-1) = 1 - 1/2 + 1/3 - 1/4 = 7/12 \).
495. Which converges to 3? I. \( \sum_{n=0}^\infty \frac{2}{3^n} \) II. \( 1 + \ln 3 + \frac{(\ln 3)^2}{2} + \cdots \) III. \( \sum_{n=0}^\infty \frac{3n+1}{n-8} \)
答案： ( C )
I: geometric \( 2/(1-1/3) = 3 \). II: \( e^{\ln 3} = 3 \). III: term \( \not\to 0 \), diverges.
496A. 496. (A) Show that \( \sum_{n=1}^\infty \frac{n^n}{(3n)!} \) is convergent.
Ratio test: \( \left|\frac{a_{n+1}}{a_n}\right| = \frac{1}{3}(1+1/n)^n \frac{1}{(3n+2)(3n+1)} \to 0 < 1 \); series converges.
496B. 496. (B) Use part (a) to evaluate \( \lim_{n\to\infty} \frac{n^n}{(3n)!} \).
Since the series converges, the nth term must tend to 0; \( \lim_{n\to\infty} \frac{n^n}{(3n)!} = 0 \).
497A. 497. (A) Approximate \( f(x) = 1/x^2 \) by \( T_3(x) \), the Taylor polynomial of degree 3 centered at 1.
\( f(1)=1 \), \( f'(1)=-2 \), \( f''(1)=6 \), \( f'''(1)=-24 \); \( T_3(x) = 1 - 2(x-1) + 3(x-1)^2 - 4(x-1)^3 \).
497B. 497. (B) Use Taylor's inequality to estimate the accuracy when \( 0.8 \leq x \leq 1.2 \).
\( |R_3| \leq \frac{M}{4!}|x-1|^4 \); \( f^{(4)}(x) = 120/x^6 \) max at 0.8; error \( \leq \frac{120/0.8^6}{24}(0.2)^4 \).
498A. 498. Suppose the \( N \)th partial sum of \( \sum_{n=0}^\infty a_n \) is \( s_N = N\tan(\pi/N) \). (A) Find \( a_4 \).
\( a_4 = s_4 - s_3 = 4\tan(\pi/4) - 3\tan(\pi/3) = 4 - 3\sqrt{3} \).
498B. 498. (B) Find \( \sum_{n=0}^\infty a_n \).
\( \lim_{N\to\infty} s_N = \lim_{N\to\infty} N\tan(\pi/N) = \pi \) (limit \( \pi \)).
499A. 499. \( \sum c_n x^n \) converges at \( x = -3 \) and diverges at \( x = 5 \). (A) Does \( \sum c_n \) converge?
At \( x=1 \): \( 1 \) is in \( (-3,5) \) (radius \( \geq 4 \)), so \( \sum c_n \) converges.
499B. 499. (B) Does \( \sum c_n 6^n \) converge?
\( |6| > 5 \); outside interval of convergence, so \( \sum c_n 6^n \) diverges.
499C. 499. (C) Does \( \sum c_n (-2)^n \) converge?
\( |-2| = 2 < 5 \); inside \( (-3,5) \), so \( \sum c_n (-2)^n \) converges.
499D. 499. (D) Does \( \sum (-1)^n c_n 7^n \) converge?
\( |7| > 5 \); diverges.
500A. 500. (A) Express \( \frac{\sin x}{x} \) as a power series.
\( \frac{\sin x}{x} = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n+1)!} \) for \( x \neq 0 \); 1 at \( x=0 \).
500B. 500. (B) Evaluate \( \int \frac{\sin x}{x}\,dx \) as an infinite series.
\( \int \frac{\sin x}{x}\,dx = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)(2n+1)!} + C \).
500C. 500. (C) Approximate \( \int_0^1 \frac{\sin x}{x}\,dx \) to three decimal places.
Integrate the series from (A) term-by-term from 0 to 1; partial sums give \( \approx 0.946 \).