Find S = 4/7 - 7/7² + 10/7³ - 13/7⁴.........infinity
S = 4/7 - 7/7^2 + 10/7^3 - 13/7^4
multiply both sides by -1/7, and line up the denominators
-1/7S = -4/7^2 + 7/7^3 - 10/7^4 + 13/7^5 - ....
subtract them:
(8/7)S=4/7 - 3/7^2 + 3/7^3 - 3/7^4 .....
= 4/7 - (3/7^2 - 3/7^3 + 3/7^4 - ......)
the part in the bracket is a GP with a = 3/49 and r = -1/7
sum of the terms in the bracket = (3/49) / (1 + 1/7)
= (3/49)(7/8) = 3/56
so (8/7)S = 4/7 - 3/56 = 29/56
S = (29/56)(7/8) = 29/64
29/64
To find the sum of the infinite series S = 4/7 - 7/7² + 10/7³ - 13/7⁴ + ..., we can use the formula for the sum of an infinite geometric series.
The formula is: S = a / (1 - r), where "a" is the first term and "r" is the common ratio.
In this case, the first term "a" is 4/7 and the common ratio "r" is -1/7.
Now we can substitute these values into the formula to find the sum:
S = (4/7) / (1 - (-1/7))
To simplify, we multiply the numerator and denominator of the fraction by the reciprocal of the denominator:
S = (4/7) / (1 + 1/7)
Simplifying further, we find the common denominator:
S = (4/7) / (7/7 + 1/7)
S = 4/7 / 8/7
Dividing fractions by multiplying by the reciprocal, we get:
S = (4/7) * (7/8)
S = 1/2
Therefore, the sum of the infinite series S = 4/7 - 7/7² + 10/7³ - 13/7⁴ + ... is 1/2.
To find the sum of the given infinite series, we can use the formula for the sum of an infinite geometric series. The general form of a geometric series is:
S = a/(1 - r)
Where:
S = sum of the geometric series
a = first term of the series
r = common ratio between the terms
In this case, the first term of the series is 4/7 and the common ratio is -7/7² = -1/7.
Substituting these values into the formula, we get:
S = (4/7) / (1 - (-1/7))
To simplify, we need to find a common denominator:
S = (4/7) / (1 + 1/7)
Now, let's add the fractions:
S = (4/7) / (7/7 + 1/7)
S = (4/7) / (8/7)
To divide fractions, we multiply the first fraction by the reciprocal of the second:
S = (4/7) * (7/8)
S = 28/56
Finally, we can simplify the fraction:
S = 1/2
So, the sum of the given infinite series S is equal to 1/2.