Evaluate the Summation

Problem

(∑_n=1^∞)(−)*4*(−1/2)(n−1)

Solution

1. Identify the type of series. This is a geometric series of the form (∑_n=1^∞)(a)*r(n−1) where a is the first term and r is the common ratio.

2. Determine the first term a By substituting n=1 into the expression, we find a=−4*(−1/2)(1−1)=−4*(1)=−4

3. Determine the common ratio r The base of the exponent is r=−1/2

4. Check for convergence. Since |r|=|−1/2|=1/2 and 1/2<1 the infinite geometric series converges.

5. Apply the formula for the sum of an infinite geometric series, which is S=a/(1−r)

6. Substitute the values into the formula:

S=(−4)/(1−(−1/2))

7. Simplify the denominator:

S=(−4)/3/2

8. Calculate the final result by multiplying by the reciprocal:

S=−4⋅2/3=−8/3

Final Answer

(∑_n=1^∞)(−)*4*(−1/2)(n−1)=−8/3

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