Finding the value of an infinite geometric series is to find the sum of the terms in the series.
Formula to find the sum of infinite geometric series :
where -1 < r < 1
In the formula above, a_{1} is the first term of the series and r is the common ratio.
r = second term/first term
or
r =a_{2}/a_{1}
Note :
In an infinite geometric series, if the value of r is not in the interval -1 < r < 1, then the sum does not exist.
Find the values of the following infinite geometric series :
Example 1 :
1 + 3/4 + 9/16 + 27/64 ............∞
Solution :
In the given geometric series,
a_{1} = 1
r = a_{2}/a_{1}
= (3/4)/1
= 3/4
Formula to find the sum of an infinite geometric series :
S_{∞} = a_{1}/(1 - r)
Substitute a_{1} = 1 and r = 3/4.
S_{∞}= 1/(1 - 3/4)
= 1/(1/4)
= 1(4/1)
= 4
The value of the given infinite geometric series is 4.
Example 2 :
1 + 2/3 + 4/9 + 8/27 ............∞
Solution :
In the given geometric series,
a_{1}= 1
r = a_{2}/a_{1}
= (2/3)/1
= 2/3
Formula to find the sum of an infinite geometric series :
S_{∞}= a_{1}/(1 - r)
Substitute a_{1}= 1 and r = 2/3.
S_{∞}= 1/(1 - 2/3)
= 1/(1/3)
= 1(3/1)
= 3
The value of the given infinite geometric series is 3.
Example 3 :
1 + 1/2 + 1/4 + 1/8 ............∞
Solution :
In the given geometric series,
a_{1}= 1
r = a_{2}/a_{1}
= (1/2)/1
= 1/2
Formula to find the sum of an infinite geometric series :
S_{∞}= a_{1}/(1 - r)
Substitute a_{1}= 1 and r = 1/2.
S_{∞}= 1/(1 - 1/2)
= 1/(1/2)
= 1(2/1)
= 2
The value of the given infinite geometric series is 2.
Example 4 :
1 + 3/5 + 9/25 + 27/125 ............∞
Solution :
In the given geometric series,
a_{1}= 1
r = a_{2}/a_{1}
= (3/5)/1
= 3/5
Formula to find the sum of an infinite geometric series :
S_{∞}= a_{1}/(1 - r)
Substitute a_{1}= 1 and r = 3/5.
S_{∞}= 1/(1 - 3/5)
= 1/(2/5)
= 1(5/2)
The value of the given infinite geometric series is 5/2.
Example 5 :
1 + 1/4 + 1/16 + 1/64 ............∞
Solution :
In the given geometric series,
a_{1}= 1
r = a_{2}/a_{1}
= (1/4)/1
= 1/4
Formula to find the sum of an infinite geometric series :
S_{∞}= a_{1}/(1 - r)
Substitute a_{1}= 1 and r = 1/4.
S_{∞}= 1/(1 - 1/4)
= 1/(3/4)
= 1(4/3)
The value of the given infinite geometric series is 4/3.
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