Answers:
The series converges
The sum is -8/3
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a = -4 is the first term
r = -1/2 is the common ratio
the reason why is because the expression -4(-1/2)^(n-1) is in the form a(r)^(n-1). Notice how -4 matches with the 'a'; the -1/2 matches with the r
The common ratio will determine if the infinite series converges or not.
r = -1/2 leads to |r| = |-1/2| = 1/2 = 0.5
Since |r| < 1 (in this case 0.5 < 1) is true, this means the series does converge.
Put another way, because r = -1/2 = -0.5 makes -1 < r < 1 true, the series converges.
Use the formula below to find the converging value S
S = a/(1-r)
S = -4/(1-(-1/2))
S = -4/(1+1/2)
S = -4/(2/2+1/2)
S = -4/(3/2)
S = (-4/1)/(3/2)
S = (-4/1)*(2/3)
S = (-4*2)/(1*3)
S = -8/3
The infinite series converges to -8/3
So the sum is -8/3
