Sequences

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Sequences

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.

Sequences

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25, …is the sequence of square numbers.

Sequences

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25, …is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.

Sequences

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25, …is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.

as {f(n)}n=1.

Sequences

One way to describe a sequence is to give the formula for the general term of the sequence, that is

∞

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25, …is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.

as {f(n)}n=1. We also write f(n) as an.

Sequences

One way to describe a sequence is to give the formula for the general term of the sequence, that is

∞

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25, …is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.

as {f(n)}n=1. We also write f(n) as an.

Sequences

One way to describe a sequence is to give the formula for the general term of the sequence, that is

∞

Example:A. The sequence {3n + 1}n=1 =

∞

A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.

Example:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25, …is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.

as {f(n)}n=1. We also write f(n) as an.

Sequences

One way to describe a sequence is to give the formula for the general term of the sequence, that is

∞

Example:A. The sequence {3n + 1}n=1 = {4, 7, 10, …} ∞

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …

Sequences

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …A sequence with alternating positive and negative terms is called an alternating sequence.

Sequences

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …A sequence with alternating positive and negative terms is called an alternating sequence.The formulas for alternating sequences contain the multiplier of (–1)n. which switches the sign.

Sequences

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …A sequence with alternating positive and negative terms is called an alternating sequence.The formulas for alternating sequences contain the multiplier of (–1)n. which switches the sign.Example C. The sequence {(–1)n(2n – 1)}n=1 is the alternating sequence –1, 3, –5, 7, –9, …

∞

Sequences

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …A sequence with alternating positive and negative terms is called an alternating sequence.The formulas for alternating sequences contain the multiplier of (–1)n. which switches the sign.Example C. The sequence {(–1)n(2n – 1)}n=1 is the alternating sequence –1, 3, –5, 7, –9, …

∞

D. Give the formula for the sequence , , , , ...-49

23

627

-881

Sequences

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …A sequence with alternating positive and negative terms is called an alternating sequence.The formulas for alternating sequences contain the multiplier of (–1)n. which switches the sign.Example C. The sequence {(–1)n(2n – 1)}n=1 is the alternating sequence –1, 3, –5, 7, –9, …

∞

D. Give the formula for the sequence , , , , ...-49

23

627

-881

The numerators are 2n, the denominators are 3n andthe signs alternates, so its {(-1)n-12n/3n}n=1

. ∞

Sequences

B. The sequence of odd numbers 1, 3, , …may be given as {2n – 1}, n =1, 2, ...It also may be given as {2n +1}, n = 0, 1, 2, …A sequence with alternating positive and negative terms is called an alternating sequence.The formulas for alternating sequences contain the multiplier of (–1)n. which switches the sign.Example C. The sequence {(–1)n(2n – 1)}n=1 is the alternating sequence –1, 3, –5, 7, –9, …

∞

D. Give the formula for the sequence , , , , ...-49

23

627

-881

The numerators are 2n, the denominators are 3n andthe signs alternates, so its {(-1)n-12n/3n}n=1

. ∞

Sequences

Limit of a sequence lim f(n) is just lim f(x), if it exists.

x∞x∞

Sequences

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/xx∞n∞

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x )1/x

x∞n∞ x∞

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x ) = lim eLn(x)/x1/x

x∞n∞ x∞ x∞

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x ) = lim eLn(x)/x1/x

x∞n∞ x∞ x∞

0

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x ) = lim eLn(x)/x = e0 = 11/x

x∞n∞ x∞ x∞

0

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x ) = lim eLn(x)/x = e0 = 11/x

x∞n∞ x∞ x∞

0

Hence an = { n1/n } is a convergent (CG) sequence.

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x ) = lim eLn(x)/x = e0 = 11/x

x∞n∞ x∞ x∞

0

Hence an = { n1/n } is a convergent (CG) sequence.

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

n∞ n∞

The Sandwich Theorem:If 0 ≤ an ≤ bn for all n that are sufficiently large* and that lim bn = 0, then lim an = 0.

Limit of a sequence lim f(n) is just lim f(x), if it exists. x∞x∞

Sequences

Example: Let an = { n1/n }, find lim an if it exists.x∞

lim n1/n = lim x1/x = lim eLn(x ) = lim eLn(x)/x = e0 = 11/x

x∞n∞ x∞ x∞

0

n∞ n∞* “Sufficiently large n" means that "for all the numbers that are greater than certain fixed number c".

The Sandwich Theorem:If 0 ≤ an ≤ bn for all n that are sufficiently large* and that lim bn = 0, then lim an = 0.

Hence an = { n1/n } is a convergent (CG) sequence.

We say the sequence converges (CG) if the limit is finite, we say it diverges (DG) if the limit is ±∞ or if it doesn't exist.

SequencesExample: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn,

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2n

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn.

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120.

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120. (We define 0! = 1)

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120. (We define 0! = 1)

Example: Justify that lim 1/n! = 0 as n∞

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120. (We define 0! = 1)

Example: Justify that lim 1/n! = 0 as n∞

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Compare {1/n!} with {1/n},

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120. (We define 0! = 1)

Example: Justify that lim 1/n! = 0 as n∞1/n > 1/n! > 0 for n > 3.

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Compare {1/n!} with {1/n},

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120. (We define 0! = 1)

Example: Justify that lim 1/n! = 0 as n∞1/n > 1/n! > 0 for n > 3.

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Compare {1/n!} with {1/n},Since lim 1/n = 0

Sequences

Need to find a number c so that if n > c we’ve an > bn, i.e. n2 – 10 > 2nn2 – 2n – 10 > 0(n – 5)(n + 3) > 0. From this inequality, we see that if we set c = 5 then for n > c we’ll have that an > bn. Recall for n = 0, 1, 2,.., the numbern! = n(n – 1)(n – 2)…3*2*1 is called the n factorial.For example 5! = 5*4*3*2*1 = 120. (We define 0! = 1)

Example: Justify that lim 1/n! = 0 as n∞1/n > 1/n! > 0 for n > 3.

Example: Given that {an} = {n2 – 15} and {bn} = {2n} for n = 1, 2, .. show that an > bn for sufficiently large n.

Compare {1/n!} with {1/n},Since lim 1/n = 0 so lim 1/n! = 0 as n ∞.

Sequencesn∞Example: Justify that lim 10n/n! = 0

Sequencesn∞Example: Justify that lim 10n/n! = 0

Compare bn = {1011/n} to an = {10n/n!}.

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

Compare bn = {1011/n} to an = {10n/n!}.

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0.

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!= 10 * 10….. 10*10*10*10…10

n *(n-1)….11*10* 9 * 8…..1

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!= 10 * 10….. 10*10*10*10…10

n *(n-1)….11*10* 9 * 8…..1

1

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!= 10 * 10….. 10*10*10*10…10

n *(n-1)….11*10* 9 * 8…..1

10101

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!= 10 * 10….. 10*10*10*10…10

n *(n-1)….11*10* 9 * 8…..1

10101

< 1011

n

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!= 10 * 10….. 10*10*10*10…10

n *(n-1)….11*10* 9 * 8…..1

10101

< 1011

n

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

0n∞

Sequencesn∞Example: Justify that lim 10n/n! = 0

We will show that bn > an (> 0) for sufficiently large n,

For if n > 10 we note that 10n

n!= 10 * 10….. 10*10*10*10…10

n *(n-1)….11*10* 9 * 8…..1

10101

< 1011

n

Hence for large n, 1011/n > 10n/n!, and the fact 1011/n 0 implies lim10n/n! = 0

n∞

Compare bn = {1011/n} to an = {10n/n!}.

and since that bn0 we must have that an0. Specifically we claim that bn > an for n > 10

0

n∞