The nth-Term and Integral Tests (BC)

Question 1

A student writes: "For _n=1^ 5n, the terms go to 0, so by the nth-Term Test the series converges." Is the reasoning correct?

Accepted Answer

No. A limit of 0 leaves the test inconclusive, and this series diverges as a p-series with p = 12

Answer

Yes. The test runs both ways, so a limit of 0 settles convergence as surely as a nonzero limit settles divergence

Answer

No, but only because the sum starts at n = 1; from n = 2 the same reasoning would hold

Answer

Yes, and the sum is 0, since a convergent series adds up to the limit of its own terms

Question 2

What does the nth-Term Test tell you about _n=1^ 2n^2 + 18n^2 - 2n?

Accepted Answer

It diverges, because _n a_n = 14 0

Answer

It converges, to 14, because _n a_n = 14

Answer

The test is inconclusive, because _n a_n exists and is finite

Answer

It converges, because a_n levels off at 14 instead of growing

Question 3

Apply the Integral Test to _n=1^ ne^-n^2. What is the conclusion?

Accepted Answer

It converges, because _1^ of the matching function equals 12e

Answer

It converges, because a_n 0 and terms that vanish must add to something finite

Answer

It converges, and its sum is 12e, the exact value of that same integral

Answer

The Integral Test does not apply, because -12e^-x^2 is not elementary on [1, )

Question 4

Apply the Integral Test to _n=1^ 1n. What is the conclusion?

Accepted Answer

It diverges, because _1^ of the matching function is infinite

Answer

It converges, because a_n 0 and terms that vanish must add to something finite

Answer

It converges, and its sum is 2, the value of 2x at the lower limit

Answer

The Integral Test does not apply, because 2x is not elementary on [1, )

Question 5

What does the nth-Term Test tell you about _n=1^ 7n^2 - 36n^2 - 4n?

Accepted Answer

It diverges, because _n a_n = 76 0

Answer

It converges, to 76, because _n a_n = 76

Answer

The test is inconclusive, because _n a_n exists and is finite

Answer

It converges, because a_n levels off at 76 instead of growing

Question 6

Which of these p-series CONVERGES?

Accepted Answer

_n=1^ 1n^3/2

Answer

_n=1^ 1n

Answer

_n=1^ 13n

Answer

_n=1^ 1n^9/10

Question 7

Which of these p-series CONVERGES?

Accepted Answer

_n=1^ 1n^5/4

Answer

_n=1^ 1n

Answer

_n=1^ 13n

Answer

_n=1^ 1n^9/10

Question 8

A student writes: "For _n=1^ 5n, the terms go to 0, so by the nth-Term Test the series converges." Is the reasoning correct?

Accepted Answer

No. A limit of 0 leaves the test inconclusive, and this series diverges as a p-series with p = 1

Answer

Yes. The test runs both ways, so a limit of 0 settles convergence as surely as a nonzero limit settles divergence

Answer

No, but only because the sum starts at n = 1; from n = 2 the same reasoning would hold

Answer

Yes, and the sum is 0, since a convergent series adds up to the limit of its own terms

Question 9

What does the nth-Term Test tell you about _n=1^ 6n + 39n - 41?

Accepted Answer

It diverges, because _n a_n = 23 0

Answer

It converges, to 23, because _n a_n = 23

Answer

The test is inconclusive, because _n a_n exists and is finite

Answer

It converges, because a_n levels off at 23 instead of growing

Question 10

Apply the Integral Test to _n=1^ 1n^2 + 1. What is the conclusion?

Accepted Answer

It converges, because _1^ of the matching function equals 4

Answer

It converges, because a_n 0 and terms that vanish must add to something finite

Answer

It converges, and its sum is 4, the exact value of that same integral

Answer

The Integral Test does not apply, because ^-1 x is not elementary on [1, )

Question 11

Apply the Integral Test to _n=2^ 1n(n)^2. What is the conclusion?

Accepted Answer

It converges, because _2^ of the matching function equals 12

Answer

It converges, because a_n 0 and terms that vanish must add to something finite

Answer

It converges, and its sum is 12, the exact value of that same integral

Answer

The Integral Test does not apply, because -1x is not elementary on [2, )

Question 12

A student writes: "For _n=1^ 43n, the terms go to 0, so by the nth-Term Test the series converges." Is the reasoning correct?

Accepted Answer

No. A limit of 0 leaves the test inconclusive, and this series diverges as a p-series with p = 13

Answer

Yes. The test runs both ways, so a limit of 0 settles convergence as surely as a nonzero limit settles divergence

Answer

No, but only because the sum starts at n = 1; from n = 2 the same reasoning would hold

Answer

Yes, and the sum is 0, since a convergent series adds up to the limit of its own terms

Question 13

Apply the Integral Test to _n=1^ nn^2 + 1. What is the conclusion?

Accepted Answer

It diverges, because _1^ of the matching function is infinite

Answer

It converges, because a_n 0 and terms that vanish must add to something finite

Answer

It converges, and its sum is 122, the value of 12(x^2 + 1) at the lower limit

Answer

The Integral Test does not apply, because 12(x^2 + 1) is not elementary on [1, )

Question 14

For which series are the hypotheses of the Integral Test satisfied on [1, )?

Accepted Answer

_n=1^ 1n^2 + 5

Answer

_n=1^ (-1)^nn

Answer

_n=1^ nn + 5

Answer

_n=1^ 1n - 6

Question 15

What does the nth-Term Test tell you about _n=1^ 6n^2 - 54n^2 - 6n?

Accepted Answer

It diverges, because _n a_n = 32 0

Answer

It converges, to 32, because _n a_n = 32

Answer

The test is inconclusive, because _n a_n exists and is finite

Answer

It converges, because a_n levels off at 32 instead of growing

Question 16

What does the nth-Term Test tell you about _n=1^ 6n^2 - 62n^2 - 7n?

Accepted Answer

It diverges, because _n a_n = 3 0

Answer

It converges, to 3, because _n a_n = 3

Answer

The test is inconclusive, because _n a_n exists and is finite

Answer

It converges, because a_n levels off at 3 instead of growing

Question 17

Which of these p-series CONVERGES?

Accepted Answer

_n=1^ 1n^3

Answer

_n=1^ 1n

Answer

_n=1^ 13n

Answer

_n=1^ 1n^9/10

Question 18

A student writes: "For _n=1^ 23n, the terms go to 0, so by the nth-Term Test the series converges." Is the reasoning correct?

Accepted Answer

No. A limit of 0 leaves the test inconclusive, and this series diverges as a p-series with p = 13

Answer

Yes. The test runs both ways, so a limit of 0 settles convergence as surely as a nonzero limit settles divergence

Answer

No, but only because the sum starts at n = 1; from n = 2 the same reasoning would hold

Answer

Yes, and the sum is 0, since a convergent series adds up to the limit of its own terms

Question 19

For which series are the hypotheses of the Integral Test satisfied on [1, )?

Accepted Answer

_n=1^ 1n^2 + 7

Answer

_n=1^ (-1)^nn

Answer

_n=1^ nn + 7

Answer

_n=1^ 1n - 2

Question 20

For which series are the hypotheses of the Integral Test satisfied on [1, )?

Accepted Answer

_n=1^ 1n^2 + 3

Answer

_n=1^ (-1)^nn

Answer

_n=1^ nn + 3

Answer

_n=1^ 1n - 2

Question 21

For which series are the hypotheses of the Integral Test satisfied on [1, )?

Accepted Answer

_n=1^ 1n^2 + 8

Answer

_n=1^ (-1)^nn

Answer

_n=1^ nn + 8

Answer

_n=1^ 1n - 2

Question 22

Apply the Integral Test to _n=1^ 1n^3. What is the conclusion?

Accepted Answer

It converges, because _1^ of the matching function equals 12

Answer

It converges, because a_n 0 and terms that vanish must add to something finite

Answer

It converges, and its sum is 12, the exact value of that same integral

Answer

The Integral Test does not apply, because -12x^2 is not elementary on [1, )

Question 23

What does the nth-Term Test tell you about _n=2^ 3n^2 - 12n^2 - 2n?

Accepted Answer

It diverges, because _n a_n = 32 0

Answer

It converges, to 32, because _n a_n = 32

Answer

The test is inconclusive, because _n a_n exists and is finite

Answer

It converges, because a_n levels off at 32 instead of growing

Question 24

Which of these p-series CONVERGES?

Accepted Answer

_n=1^ 1n^9/8

Answer

_n=1^ 1n

Answer

_n=1^ 13n

Answer

_n=1^ 1n^9/10

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The nth-Term and Integral Tests (BC)

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The nth-Term and Integral Tests (BC)

Ways to Play

Device Games: Students play on their phones

Blast Go

Zombie Escape

Avalanche

Maze Blast

Cave Climb

Blast Out

Mystery Auction

Tour Guide

Food Truck Rush

Tug of War

Alien Blast

Glacier Rush

Board Games: Teacher-led, projected to the class

Like what you see?

More like this

Bug Blast

BlastRace

BlastQuiz

BlastBoard

Seek & Destroy

Mystery Blast

Checkers

Texas Hold 'Em

Da Bomb

Castle Siege

Hot Potato

Market Mayhem

King of the Hill

Bank Heist

Top Corner