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The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem...

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The Comparison Test Let 0 ≤ a k b k for all k. b k k =1 con ve rge s ⇒ a k k =1 con ve rge s.
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Page 1: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

The Comparison Test

Let 0 ≤ ak ≤ bk for all k.

b

kk=1

∑ converges ⇒ ak

k=1

∑ converges.

Page 2: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

The Comparison Test

Comparison Theorem AAssume that 0 ≤ ak ≤ bk for all k.

If the series converges, then also the

series converges, and

a

kk=1

∑ ≤ bk

k=1

∑ .

b

kk=1

a

kk=1

Page 3: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

The Comparison Test

Claim

ak

k=1

∑ converges.

Observe that the partial sums Sm = a1 + a2 + … + am form an increasing sequence since ak ≥ 0 for all k.

Proof

Page 4: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

The Comparison TEST

Claim

ak

k=1

∑ converges.

The assumptions imply

b

kk=1

Observe that the sum is finite since this series converges.

Proof (cont’d)

Page 5: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

The Comparison test

Claim

ak

k=1

∑ converges.

Proof (cont’d)

akk=1

m∑

The partial sums form a

bounded increasing sequence.

a

kk=1

∑ = limm→ ∞

ak

k=1

m

Hence the limit exists and is finite.

Page 6: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

The Comparison test

Comparison Theorem B

Assume that 0 ≤ ak ≤ bk for all k.

If the series diverges, then also the

series diverges.

ak

k=1

b

kk=1

Page 7: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

The Comparison Test

Claim

bk

k=1

∑ diverges.

akk=1

∞∑

Since the series diverges, the

partial sums Sm form an unbounded set.

0 ≤S

m= a

kk=1

m

∑ ≤ bk

k=1

m

∑ .

The assumptions implyProof

Page 8: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

Claim

bk

k=1

∑ diverges.

limm→ ∞

ak

k=1

m

∑ =∞.HenceProof (cont’d)

limm→ ∞

bkk=1

m∑ =∞.

Since , also

akk=1

m∑ ≤ bkk=1

m∑THE COMPARISON TEST

Page 9: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

Example

1

2k 2 + sink( )k=1

∑Show that the series converges.

For all integer values of k, 1 < 2 + sin k < 3.

Solution

0 <1

3 ×2 k<

1

2 k 2 + sink( )<

12 k

Hence for all

integer values of k.

THE COMPARISON TEST

Page 10: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

Example

1

2k 2 + sink( )k=1

∑Show that the series converges.

The series is a

convergent geometric series

Solution (cont’d)

1

2kk=1

∞∑

Hence converges by the

Comparison Theorem A.

1

2k 2 + sink( )k=1

∑THE COMPARISON TEST

Page 11: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

Example

1

k 1−sin2 k( )k=1

∑Show that the series diverges.

0 <1k

<1

k 1 −sin2 k( )

Hence for all positive

integer values of k.

For all positive integers k, Solution

0 <1 −sin2 k <1 .THE COMPARISON TEST

Page 12: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

Mika Seppälä

Example

1

k 1−sin2 k( )k=1

∑Show that the series diverges.

Since the Harmonic Seriesdiverges, also

Solution (cont’d)

diverges by the Comparison Theorem B.

1

kk=1

∞∑

1

k 1−sin2 k( )k=1

THE COMPARISON TEST

Page 13: The Comparison Test Let 0 a k b k for all k.. Mika Seppälä The Comparison Test Comparison Theorem A Assume that 0 a k b k for all k. If the series converges,

The Comparison Test

Let 0 ≤ ak ≤ bk for all k.

b

kk=1

∑ converges ⇒ ak

k=1

∑ converges.

a

kk=1

∑ diverges ⇒ bk

k=1

∑ diverges.


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