Answer:
- The series converges
Step-by-step explanation:
Given the series:
∑ -4(-1/2)^(n-1) From n = 1 to ∞
Using the ratio test,
Let a_n = -4(-1/2)^(n-1)
a_(n+1) = -4(-1/2)^n
Now,
|a_n/a_(n+1)| = [-4(-1/2)^(n-1)] ÷ [-4(-1/2)^n]
= |(-1/2)^(-1)|
= |-2|
= 2
Suppose the series converges, the
Limit as n approaches infinity of |a_n/a_(n+1)| exist.
Lim(2) as n approaches infinity
= 2
Therefore, the series converges.
The radius of convergence is 2.
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It is possible to find the sum of the series.
Using geometric progression formula
S = a/(1-r)
Where a is the first term of the sequence, and r is the common ratio
The series is
-4, 2, 1, 1/2, ...
a = -4
r = a2/a1 = a3/a2 = ...
a2/a1 = -1/2
a3/a2 = 1/2
Since a2/a1 ≠ a3/a2, the common ratio doesnot exist, and so, the sum doesn't neither.
