Final answer:
To find the sum of the infinite series ∑=1[infinity](−1)−121/3, we can determine the general term of the series and apply the formula for the sum of an infinite geometric series. The sum of the series is 1/4.
Step-by-step explanation:
The given series is ∑=1[infinity](−1)−121/3. To find the sum of this infinite series, we can determine the general term of the series and then apply the formula for the sum of an infinite geometric series.
The general term of the series is (-1)^(n-1)/3, where n represents the term number. The series alternates between positive and negative terms, with the magnitude of the terms decreasing by a factor of 1/3.
Using the formula for the sum of an infinite geometric series, we have:
Sum = a / (1 - r), where a represents the first term and r represents the common ratio.
In this case, the first term a = (-1)^0 / 3 = 1/3 and the common ratio r = -1/3. Substituting these values into the formula, we get:
Sum = (1/3) / (1 - (-1/3)) = (1/3) / (4/3) = 1/4.