Sequences

Question 1

What is a sequence?

Answer 1

An ordered list of numbers usually denoted as {a_n}.

Question 2

What does it mean for a sequence to converge?

Answer 2

The limit as n approaches infinity of a_n exists and is a finite number L.

Question 3

What does it mean for a sequence to diverge?

Answer 3

The limit as n approaches infinity of a_n does not exist or is infinite.

Question 4

How do you use L’Hôpital’s Rule for a sequence?

Answer 4

Convert the sequence a_n to a continuous function f(x), then take the limit as x approaches infinity.

Question 5

What is the Squeeze Theorem for sequences?

Answer 5

If a_n ≤ b_n ≤ c_n and the limits of a_n and c_n both equal L, then the limit of b_n must also be L.

Question 6

If the limit of |a_n| = 0, what can you conclude about a_n?

Answer 6

The limit of a_n is also 0.

Question 7

What is a monotonic sequence?

Answer 7

A sequence that is either entirely non-increasing or entirely non-decreasing.

Question 8

Define a sequence that is ‘bounded above’.

Answer 8

There exists a number M such that a_n ≤ M for all n.

Question 9

Define a sequence that is ‘bounded below’.

Answer 9

There exists a number m such that a_n ≥ m for all n.

Question 10

What is the Monotonic Sequence Theorem?

Answer 10

Every bounded, monotonic sequence is convergent.

Question 11

Under what condition does the geometric sequence {r^n} converge?

Answer 11

It converges if -1 < r ≤ 1.

Question 12

What is the limit of {r^n} if |r| < 1?

Question 13

What is the limit of {r^n} if r = 1?

Answer 13

1

Question 14

Why does {r^n} diverge if r = -1?

Answer 14

Because the terms oscillate between -1 and 1, so the limit does not exist.

Question 15

What is a recursive sequence?

Answer 15

A sequence where each term is defined using previous terms (e.g., Fibonacci or a_{n+1} = 2a_n + 1).

Question 16

How do you find the limit L of a convergent recursive sequence?

Answer 16

Replace both a_{n+1} and a_n with ‘L’ in the recursive formula and solve for L.

Question 17

What is the limit of {1 / n^p} if p > 0?

Question 18

What is the limit of {ln(n) / n}?

Answer 18

0 (Use L’Hôpital’s Rule: 1/n divided by 1).

Question 19

What is the limit of { (1 + 1/n)^n }?

Answer 19

e

Question 20

What is the limit of { n^(1/n) }?

Answer 20

1

Question 21

What is the limit of { x^n / n! } for any real number x?

Question 22

Order these from slowest to fastest growth: n!, n^n, ln(n), n^p, b^n.

Answer 22

ln(n) < n^p < b^n < n! < n^n

Question 23

What is the value of 0! (zero factorial)?

Answer 23

1

Question 24

What is a subsequence?

Answer 24

A sequence formed by selecting terms from a larger sequence in their original relative order.

Question 25

If a sequence converges to L, what do its subsequences converge to?

Answer 25

All subsequences also converge to L.

Question 26

Define a 'bounded' sequence.

Answer 26

A sequence that is bounded both above and below.

Question 27

What is the difference between a_n and S_n?

Answer 27

a_n is a single term of a sequence; S_n is the nth partial sum of a series.

Question 28

If a sequence is non-decreasing and bounded above, does it converge?

Answer 28

Yes, by the Monotonic Sequence Theorem.

Question 29

What happens to the limit of {(-1)^n * (1/n)}?

Answer 29

It converges to 0 (by the Absolute Value Theorem).

Question 30

What happens to the limit of {(-1)^n}?

Answer 30

It diverges due to oscillation.

Question 31

How do you find the limit of a rational sequence (polynomial/polynomial)?

Answer 31

Divide the numerator and denominator by the highest power of n in the denominator.

Question 32

If the degree of the numerator is greater than the denominator, the sequence...

Answer 32

Diverges to infinity (or negative infinity).

Question 33

If the degrees of the numerator and denominator are equal, the limit is...

Answer 33

The ratio of the leading coefficients.

Question 34

What is a 'strictly increasing' sequence?

Answer 34

A sequence where a_{n+1} > a_n for all n.

Question 35

What is the 'Least Upper Bound' property?

Answer 35

A property of real numbers stating that every non-empty set bounded above has a unique smallest upper bound.

Question 36

Is every convergent sequence bounded?

Answer 36

Yes.

Question 37

Is every bounded sequence convergent?

Answer 37

No (e.g., {(-1)^n} is bounded but diverges).

Question 38

What is the limit of { (n! / (n+2)!) }?

Answer 38

0 (Simplify to 1 / [(n+2)(n+1)] first).

Question 39

How can you prove a sequence is decreasing?

Answer 39

Show that a_{n+1} - a_n < 0 or that the ratio a_{n+1}/a_n < 1 for positive terms.

Question 40

What is the limit of { cos(n) / n }?

Answer 40

0 (Use the Squeeze Theorem: -1/n ≤ cos(n)/n ≤ 1/n).