find the sequence of partial sums S1, S2, S3, S4, and S5. #1, 3
1. 1 + 1/4 + 1/9 + 1/16 + 1/25 + ...
2. 1/(2*3) + 2/(3*4) + 3/(4*5) + 4/(5*6) + 5/(6*7) + ...
3. 3 - 9/2 + 27/4 - 81/8 + 243/16 - ...
Verify that the infinite series diverges. #7, 9, 12
7. sum n=0 to inf (7/6)^n
8. sum n=0 to inf 4(-1.05)^n
9. sum n=1 to inf n/(n+1)
10. sum n=1 to inf n/(2n+3)
11. sum n=1 to inf n^2/(n^2+1)
12. sum n=1 to inf n/sqrt(n^2+1)
13. sum n=1 to inf (2^n+1)/2^(n+1)
14. sum n=1 to inf n!/2^n
verify that the infinite series diverges. #15, 19,
15. sum n=0 to inf (5/6)^n
16. sum n=1 to inf 2(-1/2)^n
17. sum n=0 to inf (0.9)^n = 1 + 0.9 + 0.81 + 0.729 + ...
18. sum n=0 to inf (-0.6)^n = 1 - 0.6 + 0.36 - 0.216 + ...
19. sum n=1 to inf 1/(n(n+1)) (Hint: Use partial fractions.)
Find the sum of the convergent series. #25, 31, 34
25. sum n=0 to inf 5(2/3)^n
26. sum n=0 to inf (-1/5)^n
27. sum n=1 to inf 4/(n(n+2))
28. sum n=1 to inf 1/((2n+1)(2n+3))
29. 8 + 6 + 9/2 + 27/8 + ...
30. 9 - 3 + 1 - 1/3 + ...
31. sum n=0 to inf (1/2^n - 1/3^n)
32. sum n=0 to inf [(0.3)^n + (0.8)^n]
33. sum n=1 to inf (sin 1)^n
34. sum n=1 to inf 1/(9n^2 + 3n - 2)