The Comparison Test...2019/01/09 Β Β· The Comparison Test Let βπ π and βπ π be...
Transcript of The Comparison Test...2019/01/09 Β Β· The Comparison Test Let βπ π and βπ π be...
The Comparison Test
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 1
The Comparison Test
Let βππ and βππ be series with positive terms and suppose
ππ β€ ππ,ππ+1 β€ ππ+1,ππ+2 β€ ππ+2,β― , a) If the βbigger seriesβ βππ converges, then
the βsmaller seriesβ βππ also converges. b) On the other hand, if the βsmaller seriesβ
βππ diverges, then the βbigger seriesβ βππ also diverges.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 2
Informal Principle #1
Suppose βπ’π is series with positive terms. Constant terms in the denominator of π’π can usually be deleted without affecting the convergence of divergence of the series.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 3
Example 1
Guess if the series converge or diverge.
a) β 12π+1
βπ=1
b) β 1πβ2
βπ=1
c) β 1
π+123
βπ=1
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 4
Example 1 (continued)
Solution:
(a) β 12π+1
βπ=1 we expect to behave like
β 12π
βπ=1 , which is a convergent geometric
series (π = 12
, π = 12)
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 5
Example 1 (continued)
(b) β 1πβ2
βπ=1 we expect to behave like
β 1π
βπ=1 , which is a divergent π-series (π = 1
2)
(c) β 1
π+123
βπ=1 we expect to behave like
β 1π3
βπ=1 , which is a convergent π-series
(π = 3)
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 6
Informal Principle #2
If a polynomial in π appears as a factor in the numerator or denominator of π’π, all but the highest power of π in the polynomial may usually be deleted without affecting the convergence of divergence behavior of the series.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 7
Example 2
Guess if the series converge or diverge.
a) β 1π3+2π
βπ=1
b) β 6π4β2π3+1π5+π2β2π
βπ=1
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 8
Example 2 (continued)
Solution:
(a) β 1π3+2π
βπ=1 we expect to behave like
β 1π3
βπ=1 , which is a convergent π-series (π = 3
2).
(b) β 6π4β2π3+1π5+π2β2π
βπ=1 we expect to behave like
β 6π4
π5βπ=1 = 6β 1
πβπ=1 , which is a constant times
the divergent harmonic series.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 9
Example 3
Use the Comparison Test to determine whether
οΏ½1
2π2 + π
β
π=1
converges or diverges. Solution:
We expect this series to behave like β 12π2
βπ=1 =
12β 1
π2βπ=1 , which is a constant times a convergent π-series
(π = 2).
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 10
Example 3 (continued)
Since we expect the series to converge, we want to find βππ such that βππ converges and
12π2 + π
β€ ππ.
Notice 1
2π2 + πβ€
12π2
for π β₯ 1.
Since β 12π2
βπ=1 converges, so does β 1
2π2+πβπ=1 by
the Comparison Test. J. Gonzalez-Zugasti, University of
Massachusetts - Lowell 11
Example 4
Use the Comparison Test to determine whether
οΏ½1
2π2 β π
β
π=1
converges or diverges. Solution:
We expect this series to behave like β 12π2
βπ=1 =
12β 1
π2βπ=1 , which is a constant times a convergent π-series
(π = 2).
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 12
Example 4 (continued)
Since we expect the series to converge, we want to find βππ such that βππ converges and
12π2 β π
β€ ππ .
Unfortunately 1
2π2 β π>
12π2
for π β₯ 1.
So β 12π2
βπ=1 cannot be our choice for βππ.
J. Gonzalez-Zugasti, University of
Massachusetts - Lowell 13
Example 4 (continued)
We want to decrease the denominator of 12π2βπ
.
If we try ππ = 12π2βπ2
= 1π2
we get 1
2π2 β πβ€
12π2 β π2
=1π2
for π β₯ 1.
Since β 1π2
βπ=1 is a convergent π-series (π = 2),
our series β 12π2βπ
βπ=1 converges by the
Comparison Test.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 14
Example 5
Use the Comparison Test to determine whether
οΏ½1
π β 14
β
π=1
converges or diverges. Solution:
We expect this series to behave like β 1π
βπ=1 , which is
the divergent harmonic series. J. Gonzalez-Zugasti, University of
Massachusetts - Lowell 15
Example 5 (continued) Since we expect the series to diverge, we want to find βππ such that βππ diverges and
ππ β€1
π β 14
.
Notice 1πβ€
1
π β 14
for π β₯ 1.
Since β 1π
βπ=1 diverges, our series β 1
πβ14
βπ=1 diverges by the
Comparison Test.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 16
Example 6 Use the Comparison Test to determine whether
οΏ½1π + 5
β
π=1
converges or diverges. Solution: We expect this series to behave like β 1
πβπ=1 , which is a
divergent π-series (π = 12).
J. Gonzalez-Zugasti, University of
Massachusetts - Lowell 17
Example 6 (continued)
Since we expect the series to diverge, we want to find βππ such that βππ diverges and
ππ β€1π + 5
.
Unfortunately, 1πβ₯
1π + 5
for π β₯ 1.
So β 1π
βπ=1 cannot be our choice for βππ.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 18
Example 6 (continued)
We want to increase the denominator of 1π+5
.
If we try ππ = 1π+ π
= 12 π
we get 1
π + π=
12 π
β€1π + 5
for π β₯ 25.
Since β 12 π
βπ=1 = 1
2β 1
πβπ=1 is a constant times a
divergent π-series (π = 12),
our series β 1π+5
βπ=1 diverges by the Comparison
Test.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 19
The Limit Comparison Test
Let βππ and βππ be series with positive terms and suppose
π = limπββ
ππππ
a) If π is finite and π > 0, then the series both converge or both diverge.
b) If π = 0 and βππ converges, then βππ converges.
c) If π = β and βππ diverges, then βππ diverges.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 20
Example 7 Use the Limit Comparison Test to determine whether
οΏ½3π3 β 2π2 + 4π5 β π3 + 2
β
π=1
converges or diverges. Solution:
We expect this series to behave likeβ 3π3
π5βπ=1 = β 3
π2βπ=1 =
3β 1π2
βπ=1 , which is a constant times a convergent π-series
(π = 2).
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 21
Example 7 (continued)
π = limπββ
3π3 β 2π2 + 4π5 β π3 + 2
3π2
= limπββ
3π3 β 2π2 + 4π5 β π3 + 2
βπ2
3
Since this is a rational function with the degree of the numerator equal to the degree of the denominator (=5), this limit is equal to the ratio of the leading coefficients.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 22
Example 7 (continued)
π = limπββ
3π3 β 2π2 + 4π5 β π3 + 2
βπ2
3=
33
= 1 > 0
So, by the Limit Comparison Test,
β 3π3β2π2+4π5βπ3+2
βπ=1 converges.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 23
Thomas Simpson (1720 - 1761) Simpson was a successful text writer and did most of his work in probability. He taught at the Royal Military Academy in Woolwich. His first articles were published in the Ladies' Diary. Later he became editor of this popular journal. Simpson's rule to approximate definite integrals was developed and used before he was born. It is another of history's beautiful quirks that one of the ablest mathematicians of the 18th century is remembered not for his own work or his textbooks but for a rule that was never his, that he never claimed, and that bears his name only because he happened to mention it in one of his books.
J. Gonzalez-Zugasti, University of Massachusetts - Lowell 24