The Handy Math Answer Book

What does it mean if a series is convergent?

Convergence of a series is related to the convergence of a sequence, but don’t confuse

them. The convergence of the sequence of partial sums (usually written as {sn}) differs

greatly from the convergence of a sequence of numbers (usually written as {xn}). For

example, the series xn(and its associated sequence of partial sums, or {sn}), is con-

vergent if and only if the sequence {sn} is convergent. Thus, the total sum of the series

is the limit of the sequence {sn}, seen as:

What is a geometric sequence?

A geometric sequence—or geometric progression—is a finite or infinite sequence of

real numbers with the ratio between two consecutive terms being constant (called the

common ratioor r); in this case, each term is the previous term multiplied by a given

number. In a geometric sequence, the formula for the nth term, in terms of the first

number and the common ratio, becomes:

ana 1 rn^1

Here the first term a 1 , ris the common ratio, and the number of terms is n. For exam-

ple, for the numbers 2, 6, 18, 54, 162, a 1 is 2, ris 3, and nis 5. The equation then becomes:

2  35 ^1 162.

nlim sn= xn

n

n

1

" 3

3

=

=

nlim" 3 sn!

216

What are arithmetic series and sequences?

A

n arithmetic series—also called arithmetic progression—is one of the sim-

pler types of series in mathematics. In such a series, each new term is the

previous number plus a given number; it is usually seen in the form of a(a

d) (a 2 d) (a 3 d) , ..., a(n1)d. An example of an arithmetic

series would be 2  6  10  14 ..., and so on, in which dis equal to 4. The

initial termis the first one in the series; the difference between each term (d,or

4 in this case) is called the common difference.

An arithmetic sequence is usually in the form of a, ad, a 2 d, a 3 d,

..., and so on, in which ais the first term and dis the constant difference

between the two successive terms throughout. An example of an arithmetic

sequence is (1, 4, 7, 10, 13 ...), in which the difference is always a constant of 3.

The notation for arithmetic sequences is:

an 1 and.