What is the sum of the infinity of series 1+7/10+(7/10)^2+(7/10)^3
The series is a geometric series with first term 1 and common ratio 7/10. The sum of an infinite geometric series with first term a and common ratio r (where |r| < 1) is given by:
sum = a/(1-r)
Plugging in the values of a and r for our series, we get:
sum = 1/(1-7/10) = 1/(3/10) = 10/3
Therefore, the sum of the infinite series 1+7/10+(7/10)^2+(7/10)^3+... is 10/3.
To find the sum of an infinite series, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r),
where:
S = sum of the series
a = first term of the series
r = common ratio of the series
In this case, the first term (a) is 1, and the common ratio (r) is 7/10.
Plugging these values into the formula, we have:
S = 1 / (1 - 7/10).
To simplify this expression, we need to find a common denominator:
S = 1 / (10/10 - 7/10)
= 1 / (3/10).
Next, we can multiply the numerator and denominator by the reciprocal of the denominator:
S = (1 * 10) / (3 * 1)
= 10 / 3.
Therefore, the sum of the infinite series 1 + 7/10 + (7/10)^2 + (7/10)^3 + ... is 10/3 or approximately 3.33.