Did you know?
Did you know that an infinite geometric series is a sum of an infinite sequence of numbers in which each term is found by multiplying the previous term by a constant ratio? In this case, the series is given by the formula E = -4(1/3)^(n-1), where n starts from 1 and goes to infinity.
To find the first four terms of the series, we substitute n = 1, 2, 3, and 4 into the formula.
Term 1: E = -4(1/3)^(1-1) = -4(1/3)^0 = -4(1) = -4
Term 2: E = -4(1/3)^(2-1) = -4(1/3)^1 = -4(1/3) = -4/3
Term 3: E = -4(1/3)^(3-1) = -4(1/3)^2 = -4(1/9) = -4/9
Term 4: E = -4(1/3)^(4-1) = -4(1/3)^3 = -4(1/27) = -4/27
To determine if the series diverges or converges, we need to examine the common ratio, which is (1/3). If the absolute value of the common ratio is less than 1 (in this case, it is 1/3 which is less than 1), then the series converges. If the absolute value of the common ratio is greater than or equal to 1, then the series diverges.
To find the sum of the series, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r), where "a" is the first term and "r" is the common ratio. In our case, a = -4 and r = 1/3.
S = -4 / (1 - 1/3) = -4 / (2/3) = -4 * (3/2) = -6
Therefore, the sum of the infinite geometric series is -6.
