9.2--Telescoping Series (day #1) 1) Find the first five terms of the sequence of partial sums: 5 + 10/3 + 20/9 + 40/27 + 80/81 + ... 2) Find the first five terms of the sequence of partial sums: ∞ 4 -1 n 5 ( 3 ) n=0 1 9.2--Telescoping Series (day #1) Verify that each infinite series converges: ∞ 3) 1 (n+3) (n+5) n=1 ∞ 4) 5 2 n + 3n + 2 n=1 2 9.2--Telescoping Series (day #1) Verify that the infinite series converges: ∞ 5) 2 2 n + 11n + 28 n=1 Textbook--p. 615 Verify that the infinite series converges: ∞ 44) 1 (2n+1)(2n+3) n=1 3 *GRAPHING CALC. SCREEN SHOTS!!! MODE seq LIST MATH sum ( LIST OPS seq ( sum (seq (equation or y with eqn in y =, x, #, #, #) start end increment 4 9.2--Geometric Series (day #2) t1 Sum = ∞ 1 - r t1 = the first term r = the common ratio Find the sum of each infinite series: 6) 32 + 16 + 8 + 4 + 2 + ... 7) 40 + 30 + 22.5 + 16.875 + ... 8) 20 + 30 + 45 + 67.5 + ... 5 9.2--Geometric Series (day #2) Determine whether each series converges or diverges: ∞ 9) n (-3/8) n=0 ∞ 10) n (1.2) n=0 ∞ 11) n-1 4 (5/6) n=1 Write each repeating decimal as a fraction, then as a geometric series: 12) .58 13) .423 6 9.2--Geometric Series (day #2) Textbook, p. 614 ∞ 32) 4 n (n+4) n=1 (a) Find the sum of the series: (b) Use a graphing utility to find the indicated partial sum Sn and complete the table: n 5 10 20 50 100 Sn 7 9.2--Geometric Series (day #2) Textbook, p. 614 ∞ 33) n-1 2 (0.9) n=1 (a) Find the sum of the series: (b) Use a graphing utility to find the indicated partial sum Sn and complete the table: n 5 10 20 50 100 Sn 8 9.2--The nth-Term Test for Divergence (day #3) The nth-Term Test for Divergence (p. 612) ∞ If lim an ≠ 0, then an diverges. n ∞ n=1 NOTE!!! ∞ If lim an = 0, then we can't tell whether an n ∞ converges or diverges. n=1 14) Verify that the infinite series diverges: ∞ 2n 3n + 4 n=1 9 9.2--The nth-Term Test for Divergence (day #3) 15) Verify that the infinite series diverges: ∞ n + 1 n2 + 3 n=1 16) Find the sum of the convergent series: ∞ n n (0.3) + 2 (0.8) n=1 10 9.2--The nth Partial Sum of a Series (day #4) 17) A ball is dropped from a height of 10 feet and begins bouncing. The height of each bounce is 4/5 the height of the previous bounce. Find the total vertical distance traveled by the ball: 90' 18) Find the common ratio of the geometric series, then write the function that gives the sum of the series. Graph the function and the partial sums S3 and S5. What do you notice? 1 - 2x + 4x2 - 8x3 + ... 11.