Consider the infinite geometric series infinity sigma n=1 -4(1/3)^n-1. In this, the lower limit of the summation notion is "n=1".
a. Write the first four terms of the series.
b. Does the series diverge or converge?
c. If the series has a sum, find the sum.
a. To find the first four terms of the series, we substitute n = 1, 2, 3, and 4 into the series formula:
When n = 1: -4(1/3)^0 = -4(1) = -4
When n = 2: -4(1/3)^1 = -4(1/3) = -4/3
When n = 3: -4(1/3)^2 = -4(1/9) = -4/9
When n = 4: -4(1/3)^3 = -4(1/27) = -4/27
Therefore, the first four terms of the series are -4, -4/3, -4/9, and -4/27.
b. To determine if the series converges or diverges, we need to find the common ratio of the geometric series. The common ratio is the factor by which each term is multiplied to get the next term. In this case, the common ratio is (-1/3).
For a geometric series to converge, the absolute value of the common ratio must be less than 1. Since the absolute value of -1/3 is less than 1 (i.e., 1/3 < 1), the series converges.
c. To find the sum of the series, we 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 (-4)
r = common ratio (-1/3)
Substitute a = -4 and r = -1/3 into the formula:
S = -4 / (1 - (-1/3))
S = -4 / (1 + 1/3)
S = -4 / (4/3)
S = -4 * 3/4
S = -12 / 4
S = -3
Therefore, the sum of the series is -3.