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Series Convergence/Divergence Flow Chart TEST FOR DIVERGENCE Does lim a =0? NO Pan Diverges n→∞ n YES p-SERIES p YES Pan Converges Does an = 1=n , n ≥ 1? YES Is p > 1? NO NO Pan Diverges P GEOMETRICSERIES ∞ a = a n=1 n 1−r Does a =arn−1, n ≥ 1? YES Is |r| < 1? YES n NO NO Pan Diverges ALTERNATING SERIES n Does an = (−1) bn or YES Is bn+1 ≤ bn & lim bn = 0? YES Pan Converges n−1 n→∞ an = (−1) bn, bn ≥ 0? NO TELESCOPING SERIES Dosubsequenttermscanceloutprevioustermsinthe Does YES Pan=s sum? May have to use partial fractions, properties YES lim sn = s n→∞ of logarithms, etc. to put into appropriate form. s finite? NO Pan Diverges NO P TAYLORSERIES ∞ a =f(x) YES n=0 n f(n)(a) n YES Is x in interval of convergence? Does an = n! (x−a) ? NO Pa Diverges n NO Try one or more of the following tests: COMPARISONTEST YES Is 0 ≤ an ≤ bn? YES Pan Converges Pick {bn}. Does Pbn converge? NO NO Is 0 ≤ bn ≤ an? YES Pan Diverges LIMIT COMPARISON TEST a ∞ Pan Converges n Pick {bn}. Does lim b = c > 0 X YES n→∞ n YES Does bn converge? c finite & an;bn > 0? n=1 NO Pan Diverges INTEGRALTEST P ∞ a Converges Does an = f(n), f(x) is contin- Z ∞ YES n=a n uous, positive & decreasing on YES Does f(x)dx converge? [a;∞)? a NO Pan Diverges RATIO TEST YES Pan Abs. Conv. a Is lim |a =a | 6= 1? Is lim n+1 <1? n→∞ n+1 n YES a n→∞ n NO Pan Diverges ROOTTEST p Pan Abs. Conv. p n YES Is lim n |a | 6= 1? YES Is lim |an| < 1? n→∞ n n→∞ NO Pan Diverges Problems 1-38 from Stewart’s Calculus, page 784 ∞ 2 ∞ ∞ 1. X n −1 14. Xsin(n) 27. X kln(k) 2 3 n=1 n +n n=1 k=1 (k +1) ∞ ∞ Xn−1 X n! ∞ 1=n 2. n2 +n 15. 2·5·8·····(3n+2) 28. Xe 2 n=1 n=0 n=1 n ∞ ∞ 2 X 1 Xn +1 ∞ −1 3. n=1 n2 +n 16. n=1 n3 +1 29. Xtan√(n) n=1 n n ∞ ∞ X n−1n−1 X n1=n ∞ √ 4. (−1) n2 +n 17. (−1) 2 X j j n=1 n=1 30. (−1) j +5 j=1 ∞ n+1 ∞ n−1 X(−3) X(−1) 5. 3n 18. √ ∞ k n=1 2 n=2 n−1 31. X k5 k 3 +4 ∞ ∞ n k=1 X 3n X nln(n) 6. 1+8n 19. (−1) √ ∞ n n n=1 n=1 32. X(2n) n2n ∞ ∞ 7. X p1 20. Xk+5 n=1 n ln(n) 5k ∞ n=2 k=1 Xsin(1=n) ∞ k ∞ 33. √ X 2n n 8. 2 k! 21. X (−2) n=1 (k +2)! nn k=1 n=1 ∞ ∞ ∞ √ 34. X 1 2 9. Xk2e−k 22. X n2 −1 n=1 n+ncos (n) n3 +2n2+5 k=1 n=1 2 ∞ n ∞ ∞ X n X 3 X 35. 10. n2e−n 23. tan(1=n) n+1 n=1 n=1 n=1 ∞ ∞ n+1 ∞ X 1 11. X(−1) 24. X cos(n=2) 36. ln(n) nln(n) n2 +4n n=2 (ln(n)) n=2 n=1 ∞ ∞ ∞ X√ X n n Xn! n n 12. (−1) 2 25. 2 37. ( 2−1) n +25 en n=1 n=1 n=1 ∞ n 2 ∞ 2 ∞ X3n Xn +1 X√ 13. 26. 38. ( n 2 − 1) n! 5n n=1 n=1 n=1
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