
Sequences and
Series



Series

The sum of an infinite geometric
sequence, infinite geometric series 
Converting recurring decimals
(infinite decimals) to fraction






The sum of an infinite geometric
sequence, infinite geometric series 
An
infinite geometric series converges (has a finite sum even when n
is infinitely large) only
if the absolute ratio of successive terms is less than 1 that is,
if
1
< r <
1. 
The
sum of an infinite geometric series can be calculated as the value that
the finite sum formula takes (approaches) as number of terms n
tends to infinity, 
first
rewrite
S_{n},
into 



so
that 

since
 r
 < 1, then r^{n}
®
0
as n
® oo 

thus, 

the sum of an infinite converging
geometric series.



Converting recurring decimals
(infinite decimals) to fraction

Recurring
or repeating decimal is a rational number (fraction) whose
representation as a decimal contains a pattern of digits that repeats
indefinitely after decimal point. 
The
decimals that start their recurring cycle immediately after the decimal
point are called purely
recurring decimals. 
Purely
recurring decimals convert to an irreducible fraction whose prime
factors in the denominator can only be the prime numbers other than 2 or
5, i.e., the prime numbers from the sequence {3, 7, 11, 13,17, 19,
. . }. 
The
decimals that have some extra digits before the repeating sequence of
digits are called the mixed
recurring decimals. 
The
repeating sequence may consist of just one digit or of any finite number
of digits. The number of digits in the repeating pattern is called the period. 
Mixed
recurring decimals convert to an irreducible fraction whose denominator
is a product of 2's and/or 5's besides the prime numbers from the
sequence {3, 7, 11, 13,17, 19,
. . .
}. 
All
recurring decimals are infinite decimals. 

Converting
purely recurring decimals to fraction

Example:
Convert the
purely recurring decimal


to fraction.


Solution: Given
decimal we can write as the sum of the infinite
converging geometric series


Notice
that, when converting a purely recurring decimal less than one to
fraction, write the repeating digits to the numerator, and to the
denominator of the equivalent fraction write as much 9's as is the
number of digits in the repeating pattern. 
Thus,
for example: 



Converting mixed
recurring decimals to fraction

Example:
Convert the
mixed recurring decimal


to fraction.


Solution: Given
decimal we can write as the sum of 0.3
and the infinite
converging geometric series,


Since
the repeating pattern is the infinite
converging geometric series whose ratio of successive
terms is less than 1, i.e., r
= 0.01 then we use the formula for
the sum of the infinite geometric series S_{oo}
= a_{1} / (1  r), 

Notice
that, when converting the mixed recurring decimal less than one to fraction, write the difference between the number formed by the entire sequence of digits, including the digits of the recurring part, and the number formed only by the digits of the nonrecurring
pattern to its numerator. To the denominator of the equivalent fraction write as much 9’s as is the number of digits in the repeating
pattern and add as much 0’s as is the number of digits in the nonrecurring pattern. 
Thus,
for example: 










Intermediate
algebra contents 



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