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Evaluate the sum of the infinite series (1)/(2) ((1)/(3)+(1)/(4))-(1)/(4) ((1)/(3²)+(1)/(4²))+(1)/(6) ((1)/(3³)+(1)/(4³))-⋯.

1. (1)/(2)log 2
2. log(3/5)
3. log(5/3)
4. (1)/(2)log(5/3)

Correct answer: (1)/(2)log(5/3)

Solution

The nth term is (-1)^(n+1)/(2n)[(1/3)^n+(1/4)^n]. Using sum (-1)^(n+1)x^n/n = ln(1+x), the total is (1/2)[ln(4/3)+ln(5/4)] = (1/2)ln(5/3).

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