Determine if the series converges absolutely, converges conditionally, or diverges.Justify. 9)
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Q
n=1
(-1)n (3/4. - 4/n)^n
To determine the convergence of this series, we will analyze the limit of the terms.
Consider the term (-1)^n (3/4. - 4/n)^n as n approaches infinity.
First, let's simplify the expression inside the parentheses: (3/4. - 4/n)^n
As n approaches infinity, 4/n approaches 0, so (3/4. - 4/n) approaches (3/4).
Therefore, the term (-1)^n (3/4. - 4/n)^n can be simplified as (-1)^n (3/4)^n.
Now, let's consider the absolute value of this term: |(-1)^n (3/4)^n| = |(-1)^n| * |(3/4)^n| = 1 * (3/4)^n = (3/4)^n.
We can see that the term (3/4)^n approaches 0 as n approaches infinity since 0 < 3/4 < 1.
So, the absolute value of the term tends to 0 as n approaches infinity, which means the series converges absolutely.
Therefore, the series converges absolutely.