Sequences
In simple words sequence is a list of numbers written in definite order: a1 , a2 , … , an . a1 is first
term, a2 is second term, and, in general, an is n-th term. We deal with infinite sequences, so each
term an will have a successor an+1 .
Notice that for every positive integer n there is a corresponding number an and so a sequence
can be defined as a function whose domain is the set of positive integers. But we usually write an
instead of the function notation f(n) for the value of the function at the number n.
∞
The sequence {a1 , a2 , … , an } is also denoted by {an } or {an }n=1 .
If sequence is given by formula an = f (n) then we calculate an by calculating f (n). For example,
1 1
if an = n+1
then a3 =
3+1
= 14 .
Example 1. Some sequences can be defined by giving a formula for the n-th term. In the
following examples we give three descriptions of the sequence: one by using the preceding
notation, another by using the defining formula, and a third by writing out the terms of the
sequence. Notice that n doesn't have to start at 1.
∞
a. { n2n+1 } , an = n2n+1 , { 12 , 25 , 10
3 4
n=1
, 17 , … , n2n+1 , …}
n n n
b. {(−1) 2n }, an = (−1) 2n , {−2, 4, −8, 16, … , (−1) 2n , …}
∞
1 1 1 1 1 1
c. { n−2 } , an = , n > 2, {1, , , 2
, … , …}
n−2 2 3 n−2
n=3
πn ∞
d. {sin ( 4 )}n=0 ,
an = sin ( πn
4 ), n ≥ 0, {0,
1
2
, 1, 12 , 0, … , sin ( πn
4 ), … }
Note, that there are sequences that don't have a simple defining equation.
Example 2.
a. Sequence {pn } where pn is population of USA as of January 1 in the year n.
b. If an is n-th decimal digit of number π then {an }
= {1, 4, 1, 5, 9, 2, 6, 5, …}.
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, c. The Fibonacci sequence {f n } is defined recursively as f 1
= 1, f 2 = 1,f n = f n−1 + f n−2 , n ≥ 3.
In other words, n-th member is sum of two previous. So, {f n } =
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …}.
n n 1 n 1
Now, consider the sequence an = n+1
. Since n+1 =1− n+1
then
1− n+1
= n+1
. We see
that this difference can be made as small as we like by taking n sufficiently large. We indicate it
n
by writing limn→∞ n+1 = 1.
Definition. A sequence {an } has the limit L and we write limn→∞ an
= L or an → L as L →
∞ if we can make the terms an as close to L as we like by taking n sufficiently large. If
limn→∞ an exists, we say the sequence converges (or is convergent). Otherwise, we say the
sequence diverges (or is divergent).
Notice that the following definition of the limit of a sequence is very similar to the definition of a
limit of a function at infinity.
If you compare Definition with definition of a limit of a function at infinity you will see that the only
difference between limx→∞ f (x)
= L and limn→∞ an = L is that n is required to be an integer.
Thus, we have the following theorem.
Theorem 1. If limx→∞ f (x)
= L and f (n) = an when n is an integer, then limn→∞ an = L.
In particular, since limx→∞ x1r
= 0 when r > 0, we have that limn→∞
1
nr
= 0 when r > 0.
If an becomes large as n grows we use notation limn→∞ an
= ∞. In this case sequence {an } is
divergent, but in a special way. We say that {an } diverges to ∞.
The Limits Laws also hold for the limits of sequences and their proof are similar.
Limit Laws for Convergent Sequences
1. If {an } and {bn } are convergent sequences and c is arbitrary constant then;
2. limn→∞ (an + bn ) = limn→∞ an + limn→∞ bn
3. limn→∞ (an − bn ) = limn→∞ an − limn→∞ bn
4. limn→∞ c = c
5. limn→∞ can = c limn→∞ an
This study source was downloaded by 100000898062787 from CourseHero.com on 09-29-2025 04:47:44 GMT -05:00
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In simple words sequence is a list of numbers written in definite order: a1 , a2 , … , an . a1 is first
term, a2 is second term, and, in general, an is n-th term. We deal with infinite sequences, so each
term an will have a successor an+1 .
Notice that for every positive integer n there is a corresponding number an and so a sequence
can be defined as a function whose domain is the set of positive integers. But we usually write an
instead of the function notation f(n) for the value of the function at the number n.
∞
The sequence {a1 , a2 , … , an } is also denoted by {an } or {an }n=1 .
If sequence is given by formula an = f (n) then we calculate an by calculating f (n). For example,
1 1
if an = n+1
then a3 =
3+1
= 14 .
Example 1. Some sequences can be defined by giving a formula for the n-th term. In the
following examples we give three descriptions of the sequence: one by using the preceding
notation, another by using the defining formula, and a third by writing out the terms of the
sequence. Notice that n doesn't have to start at 1.
∞
a. { n2n+1 } , an = n2n+1 , { 12 , 25 , 10
3 4
n=1
, 17 , … , n2n+1 , …}
n n n
b. {(−1) 2n }, an = (−1) 2n , {−2, 4, −8, 16, … , (−1) 2n , …}
∞
1 1 1 1 1 1
c. { n−2 } , an = , n > 2, {1, , , 2
, … , …}
n−2 2 3 n−2
n=3
πn ∞
d. {sin ( 4 )}n=0 ,
an = sin ( πn
4 ), n ≥ 0, {0,
1
2
, 1, 12 , 0, … , sin ( πn
4 ), … }
Note, that there are sequences that don't have a simple defining equation.
Example 2.
a. Sequence {pn } where pn is population of USA as of January 1 in the year n.
b. If an is n-th decimal digit of number π then {an }
= {1, 4, 1, 5, 9, 2, 6, 5, …}.
This study source was downloaded by 100000898062787 from CourseHero.com on 09-29-2025 04:47:44 GMT -05:00
https://www.coursehero.com/file/250856122/sequencespdf/
, c. The Fibonacci sequence {f n } is defined recursively as f 1
= 1, f 2 = 1,f n = f n−1 + f n−2 , n ≥ 3.
In other words, n-th member is sum of two previous. So, {f n } =
{1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …}.
n n 1 n 1
Now, consider the sequence an = n+1
. Since n+1 =1− n+1
then
1− n+1
= n+1
. We see
that this difference can be made as small as we like by taking n sufficiently large. We indicate it
n
by writing limn→∞ n+1 = 1.
Definition. A sequence {an } has the limit L and we write limn→∞ an
= L or an → L as L →
∞ if we can make the terms an as close to L as we like by taking n sufficiently large. If
limn→∞ an exists, we say the sequence converges (or is convergent). Otherwise, we say the
sequence diverges (or is divergent).
Notice that the following definition of the limit of a sequence is very similar to the definition of a
limit of a function at infinity.
If you compare Definition with definition of a limit of a function at infinity you will see that the only
difference between limx→∞ f (x)
= L and limn→∞ an = L is that n is required to be an integer.
Thus, we have the following theorem.
Theorem 1. If limx→∞ f (x)
= L and f (n) = an when n is an integer, then limn→∞ an = L.
In particular, since limx→∞ x1r
= 0 when r > 0, we have that limn→∞
1
nr
= 0 when r > 0.
If an becomes large as n grows we use notation limn→∞ an
= ∞. In this case sequence {an } is
divergent, but in a special way. We say that {an } diverges to ∞.
The Limits Laws also hold for the limits of sequences and their proof are similar.
Limit Laws for Convergent Sequences
1. If {an } and {bn } are convergent sequences and c is arbitrary constant then;
2. limn→∞ (an + bn ) = limn→∞ an + limn→∞ bn
3. limn→∞ (an − bn ) = limn→∞ an − limn→∞ bn
4. limn→∞ c = c
5. limn→∞ can = c limn→∞ an
This study source was downloaded by 100000898062787 from CourseHero.com on 09-29-2025 04:47:44 GMT -05:00
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