Math 250A Practice Questions for Test 3

Math 250A Practice Questions for Test 3 Page 2 of 2Answers: (a) S ≈ 1.55 with 0 < R 10 ≤ 0.1 (b) 200 terms10. Estimate the sum of the seriescorrect to three decimals place.∞∑ (−1) nn=01Answer:0! − 1 1! + 1 2! − · · · + 1 6!≈ 0.368, and we have |error| ≤ 0.0002. The exact value of this series ise −1 .11. Test for convergence or divergence the following series. Identify the test used and make sure all appropriatehypotheses are satisfied.(a)(e)(i)∞∑n=1∞∑n=1∞∑n=1ne −n2(b)∞∑n=1( ) n 3n(f)4n + 1e −n2(j)∞∑n=1√ n + 34n 2 + 1∞∑n=13n + 25n 2 + 1(c)(−1) n+1 n!1 · 3 · 5 · · · (2n − 1)∞∑n=1(g)n!cos ( )1n∞∑n=2(k)1n ln n∞∑n=1(d)cos(n)n 3∞∑n=0(h)(−1) n√ n + 1Answers:(a) Converges. Use the integral test.(b) Converges. Use limit comparison with a p-series (p = 3/2).(c) Diverges. Use the nth term test.(d) Converges. Use the alternating series test.(e) Converges. Use the root test.(f) Diverges. Use limit comparison with the harmonic series.(g) Diverges. Use the integral test.(h) Converges. Use the ratio test.(i) Converges. Use direct comparion with a geometric series with r = e −1 .(j) Converges. Use the ratio test.(k) Converges absolutely. Use direct comparison with a p-series. (p = 3).(l) Diverges. Divergent geometric series.12. Find the interval of convergence ofAnswer: [−1, 3], i.e., −1 ≤ x ≤ 3.∞∑n=1(x − 1) nn 2 · 2 n .13. Find the first four nonzero terms of the Maclaurin series of the following.(a) xe −2x (b) cos 2 x (c) sin x cos x (d) e −x cos x (e) 3√ 1 + x 2Answers:(a) x − 2 x 2 + 2 x 3 − 4 3 x4 + · · ·(b) 1 − x 2 + 1 3 x4 − 2 45 x6 + · · ·(c) x − 2 3 x3 + 2 15 x5 − 4315 x7 + · · ·(d) 1 − x + 1 3 x3 − 1 6 x4 + · · ·(e) 1 + 1 3 x2 − 1 9 x4 + 5 81 x6 + · · ·14. Use three nonzero terms of an appropriate series to approximate the following integrals. Find an upperboundfor the error.∫ 1∫ 0.5 √(a) cos(x 2 3) dx (b) 1 + x2 dx0Answers: (a) 0.9046 with |error| < 10 −40∞∑n=1(l)1n!∞∑n=03 n+1e n(b) 0.51319 with |error| < 6.9 × 10 −5