find the sum of the series
2/3+2/9+2/27+2/81+...
Change each fraction to have a common denominator of 81
then just add them up
a = 2/3
r = 1/3
S = a/(1-r) = 1
missed that nasty "..."
To find the sum of the series 2/3 + 2/9 + 2/27 + 2/81 + ..., we can use the formula for the sum of an infinite geometric series.
In a geometric series, each term is found by multiplying the previous term by a constant ratio. In this case, the constant ratio is 1/3, since each term is one-third the value of the previous term.
The formula for the sum of an infinite geometric series is:
Sum = a / (1 - r)
Where:
- a is the first term of the series
- r is the common ratio (the constant ratio between terms)
In our case, the first term (a) is 2/3 and the common ratio (r) is 1/3.
Plugging these values into the formula, we have:
Sum = (2/3) / (1 - 1/3)
To simplify the denominator, we need to find a common denominator for 1 and 3, which is 3. So the equation becomes:
Sum = (2/3) / (3/3 - 1/3)
Simplifying further, we have:
Sum = (2/3) / (2/3)
Now we can divide the numerator by the denominator:
Sum = (2/3) * (3/2)
Multiplying, we get:
Sum = 6/6
And simplifying, we get:
Sum = 1
Therefore, the sum of the series 2/3 + 2/9 + 2/27 + 2/81 + ... is 1.