BC Calculus Series Convergence/Divergence B Notesheet Name: Direct β¦Β Β· Direct Comparison Test...
Transcript of BC Calculus Series Convergence/Divergence B Notesheet Name: Direct β¦Β Β· Direct Comparison Test...
BC Calculus Series Convergence/Divergence B Notesheet Name: _________________________________
Direct Comparison Test (DCT) If ππ β₯ 0 and ππ β₯ 0, If β ππ
βπ=1 converges and 0 β€ ππ β€ ππ, then β ππ
βπ=1 converges.
If β ππ
βπ=1 diverges and 0 β€ ππ β€ ππ, then β ππ
βπ=1 diverges.
Note: You must state/show the inequality when stating the conclusion of this test.
Example 1 Determine whether the following series converge or diverge.
a) β
π3
π3 + 1
β
π=1
b) β
1
π3
β
π=1
c) β
1
3π + 2
β
π=1
d) β
1
βπ β 1
β
π=4
e) β
|cos π|
2π
β
π=1
f)
β1
π4 β 10
β
π=2
Limit Comparison Test (LCT)
If ππ β₯ 0 and ππ β₯ 0, and limπββ
ππ
ππ= πΏ or lim
πββ
ππ
ππ= πΏ, where πΏ is both finite and positive, then the two series
β ππ
β
π=1
ππ β ππ
β
π=1
either both converge or both diverge. Note: You must show the limit when stating the conclusion of this test.
Example 2 Determine whether the following series converge or diverge.
a) β
1
3π2 β 4π + 5
β
π=1
b) β
π4
4π5 β π3 + 7
β
π=1
c) β
1
π3 β 2
β
π=2
d) β
1
β3π β 2
β
π=1
Ratio Test Let β ππ
βπ=1 be a series of nonzero terms.
β ππβπ=1 converges if lim
πββ|
ππ+1
ππ| < 1
β ππβπ=1 diverges if lim
πββ|
ππ+1
ππ| > 1
The ratio test is inconclusive if limπββ
|ππ+1
ππ| = 1
Example 3 Determine whether the following series converge or diverge.
a) β
2π
π!
β
π=1
b) β
π2(3π + 1)
2π
β
π=1
c) β
(π + 1)!
3π
β
π=1
d) β
3πβ1
π β 2π
β
π=1
Root Test Let β ππ
βπ=1 be a series of nonzero terms.
β ππβπ=1 converges if lim
πβββ|ππ|π
< 1
β ππβπ=1 diverges if lim
πβββ|ππ|π
> 1
The root test is inconclusive if limπββ
β|ππ|π= 1
Example 4 Determine whether the following series converge or diverge.
a) β
π2π
ππ
β
π=1
b) β (
3π + 4
2π)
πβ
π=1