Sequences and Series http://www.sparknotes.com/math/precalc/sequencesandseries/summary.html
A sequence is a special kind of function whose domain is the positive integers. The range of a sequence is the
collection of terms that make up the sequence. Just as the word sequence implies, the order of the terms in a sequence is
important. The first term of a sequence, for example, is found by taking the value of the function at 1 ; the second term is
the value of the function at 2 , and so on. Consider the sequence f (x) = x . The terms of the sequence, denoted a1, a2,
a3,…, an are 1, 2, 3,…, n . When working with sequences, instead of using function notation to express the formula of the
function, a formula is the following form is used: an = n . This is the same sequence as above, but the conventional n is
used to denote an integer, since only integers are in the domain of sequences. Two important categories of sequences
are arithmetic sequences, and geometric sequences. Both are examples of a recursive sequence--a sequence in
which each term (besides the first) depends on the previous term. Both of these types of sequences will be discussed.
When the terms of a sequence are summed, the result is called a series. Some series increase without bound as n
increases, but others approach a limit. Both types of series will be studied in the following sections. There are also certain
formulas for calculating the limits of series that we'll learn. The study of series is an important part of calculus, and it all
starts with sequences.
Terms and Formulae
Terms
Arithmetic Sequence - A sequence in which each term is a constant amount greater or less than the previous term. In this type of
sequence, an+1 = an + d , where d is a constant.
Example: 15,18,21,24, … is infinite arithmetic series with a 1 = 15 and common difference d = 3
Common Ratio - In a geometric sequence, the ratio r between each term and the previous term.
Example: 10,20,40,80, … is geometric with a common ratio r = 20/10=40/20=80/40 = 2
Convergent Series - A series whose limit as n→∞ is a real number.
Example: 0.3 + 0.03 + 0.003 + 0.0003 + … is infinite geometric converging with r = 1/10=0.1
So this sum increases but it approaches a sum limit (since additional terms get smaller and smaller)
… this limit sum is known to you since you recognize this sum as the decimal 0.3 repeating (
) which = 1/3
Divergent Series - A series whose limit as n→∞ is either ∞ or - ∞ .
Example: 3 + 30 + 300 + 3000 + … is infinite geometric diverging with r = 10
So this sum increases but has no sum limit (here ∞ ) since sum is positive &the additional terms get larger …
Explicit Formula - A formula for the n th term of a sequence of the form an = some function of n .
Example: 3, 5, 7, 9, … an is an Arithmetic sequence (a1=3 with d=2) where an = a1 +d(n-1) = 3 + 2(n-1) = 3 + 2n – 2 = 2n +1
Note: If you graph (n, an) for an arithmetic sequence then you will see points on a line (since n = 1,2,3…)
Example: 2, 6, 18, 54, … an is a Geometric sequence (a=2 with r=3) where an = a1(r)(n-1) = 2(6)(n-1) for n= 1,2,3…
Finite Sequence - A sequence which is defined only for positive integers less than or equal to a certain given integer.
Example: 5,10,20,40 is a finite (only 4 terms) geometric sequence (a1 = 5 with r =2) so an = 5(2)(n-1) for n = {1,2,3,4}
Finite Series - A series which is defined only for positive integers less than or equal to a certain given integer.
Example: 5+10+20+40 is a Finite Geometric Series … the sum formula will be defined for n = {1,2,3,4} = {pos. integers < 4}
Geometric Sequence - A sequence in which the ratio between each term and the previous term is a constant ratio.
Example: 64,32,16,8,4 is a finite (only 5 terms) geometric sequence with a common ratio 32/64=16/32=8/16=4/8 so r = 0.5
(all of these reduce to r =1/2 or r= 0.5)
𝑛
Index of Summation - The variable in the subscript of Sigma Σ. For ∑𝑖=1(𝑎𝑖 ), i is called the index of summation.
Example: In the finite arithmetic series given by ∑5𝑘=1(𝑎𝑘 ) = ∑5𝑘=1(6 + 5k), what is the index of summation? Answer: k
Infinite Sequence - A sequence which is defined for all positive integers.
Example: {4,6,8,10,12, …, an = 4+2(n-1)=4+2n-2=2n+2, … }
Infinite Series - A series which is defined for all positive integers.
Example: 5 + 7 + 9 + 11 + …
is an infinite arithmetic sequence (defined for n = 1,2,3,… ∞)
Recursive Sequence - A sequence in which a general term is defined as a function of one or more of the preceding terms. A
sequence is typically defined recursively by giving the first term, and the formula for any term an+1 after the first term.
Example: Famous Fibonacci Rabbits Sequence given recursively: Let F 1 = 1 and F2 = 1, then for n>3 let Fn = Fn-2 + Fn-1
yields 1, 1, 1+1=2, 1+2=3, 2+3=5, 3+5=8, 5+8=13, …
Sequence - A function which is defined for positive integers.
Example finite sequence: 5,10,15,20 defined for n= {1,2,3,4}
Example infinite sequence: 5,10,15,20,… defined for n = { 1,2,3,4,…∞}
Series - A sequence in which the terms are summed, not just listed.
Example finite series: 3+6+9+12
Example infinite series: 0.3+0.03+0.003+ 0.0003 + …
Summation Notation ∑𝑛𝑖=1 𝑎𝑖 = a1 + a2 + a3 + a4 + ... + an . The symbol Σ and its subscript (bottom) and superscript (top) are the
components of summation notation.
Term - An element in the range of a sequence. A sequence is rarely represented by ordered pairs, but instead by a list of its terms.
Example: 5,7,9,11 we see the elements a1 = 5, a2 = 7, a3 = 9 and a4 =11 *ordered pairs (1,5), (2,7), (3,9), and (4,11)
Some Formulas:
For a converging geometric sequence, one where -1 < r < 1 , the limit of the
Limit of an Infinite Converging
Geometric Series
infinite geometric series
(a1 r(n-1))=
.
The sum of the first n terms of the arithmetic sequence is Sn = n(
Sum of a Finite Arithmetic Sequence
Note: alternative version Sn = na1 +
(dn - d ) , where d=common difference.
1−𝑟 𝑛
Sum of a Finite Geometric Sequence
)
The sum of the first n terms is Sn = a1(
1−𝑟
).
General Sequences and Series
Terms and