Series Convergence/Divergence Flow Chart
TEST FOR DIVERGENCE Does limn→∞ an = 0? YES p-SERIES Does an = 1/np , n ≥ 1? NO GEOMETRIC SERIES Does an = arn−1 , n ≥ 1? NO ALTERNATING SERIES Does an = (−1)n bn or an = (−1)n−1 bn , bn ≥ 0? NO TELESCOPING SERIES Do subsequent terms cancel out previous terms in the sum? May have to use partial fractions, properties of logarithms, etc. to put into appropriate form. NO TAYLOR SERIES Does an = f (n)

NO

an Diverges an Converges an Diverges
∞ n=1

YES

Is p > 1?

YES NO

YES

Is |r| < 1?

YES NO

an =

a 1−r

an Diverges

YES

Is bn+1 ≤ bn & lim bn = 0? n→∞ YES

an Converges

YES

Does lim sn = s n→∞ s finite?

YES NO

an = s an Diverges

(a) n! (x

− a) ?

n

YES

Is x in interval of convergence?

YES NO

∞ n=0

an = f (x)

an Diverges

NO Try one or more of the following tests: Is 0 ≤ an ≤ bn ? NO NO Is 0 ≤ bn ≤ an ?
∞

COMPARISON TEST Pick {bn }. Does bn converge?

YES

YES

an Converges

YES

an Diverges an Converges an Diverges
∞ n=a

LIMIT COMPARISON TEST n Pick {bn }. Does lim an = c > 0 n→∞ b c finite & an , bn > 0?

YES

Does n=1 bn converge?

YES NO

INTEGRAL TEST Does an = f (n), f (x) is continuous, positive & decreasing on [a, ∞)? RATIO TEST Is limn→∞ |an+1 /an | = 1?

∞

YES

Does a f (x)dx converge?

YES NO

an Converges

an Diverges an Abs. Conv. an Diverges an Abs. Conv. an Diverges

YES

Is

lim an+1 n→∞ an

< 1?

YES NO

ROOT TEST Is limn→∞ n |an | = 1?

YES

Is lim

n

n→∞

|an | < 1?

YES NO

Problems 1-38 from Stewart’s Calculus, page 784

1.

n2 − 1 n2 + n n=1 n−1 n2 + n n=1 1 2+n n n=1
∞ ∞ ∞

∞

∞

∞

14. n=1 ∞

sin(n) n! 2 · 5 · 8 · · · · · (3n + 2) n=0 n2 + 1 n3 + 1 n=1
∞ ∞

27. k=1 ∞

k ln(k) (k + 1)3

2.

15.

28.

e1/n n2 n=1 tan−1 (n) √ n n n=1
∞ j ∞

3.

16. n−1 n2 + n

29.

4.
∞

(−1)n−1 n=1 17.
∞

(−1)n 21/n n=1 5.

(−3)n+1 23n n=1
∞

18. n (−1)n−1 √ n−1 n=2 ln(n) (−1)n √ n n=1
∞ ∞

√ j (−1) 30. j+5 j=1
∞

31. k=1 ∞

6. n=1 ∞

3n 1 + 8n 1 ln(n)

3k

5k + 4k

19.

32.

7. n=2 ∞

n

20. k=1 ∞

k+5 5k 33.

(2n)n n2n n=1 sin(1/n) √ n n=1 1 n + n cos2 (n) n=1
∞ ∞ ∞

8. k=1 ∞

2k k! (k + 2)! k 2 e−k

21.

(−2)2n nn n=1

9. k=1 ∞

√ ∞ n2 − 1 22. n3 + 2n2 + 5 n=1
∞
3

34.

10. n=1 ∞

n2 e−n

23. n=1 ∞

tan(1/n)

35. n=1 ∞

n n+1

n2

(−1)n+1 11. n ln(n) n=2 n 12. (−1) 2 n + 25 n=1 n ∞

cos(n/2) 24. n2 + 4n n=1 n! 25. en2 n=1 26. n2 + 1 5n n=1
∞ ∞

36.

1 (ln(n))ln(n) n=2
∞

37.
∞

n=1

√ n ( 2 − 1)n

13.

3 n n2 n! n=1
∞

38.

n=1

√ n ( 2 − 1)