Final answer:
To estimate the magnitude of error involved in using the sum of the first four terms to approximate the sum of the entire series, we need to find the sum of the entire series and compare it with the sum of the first four terms. The magnitude of the error is approximately 0.648.
Step-by-step explanation:
To estimate the magnitude of using the sum of the first four terms to approximate the sum of the entire series, we need to find the sum of the entire series and compare it with the sum of the first four terms. The given series is -1^(n+1)*(1)/(6^(n)), where n starts from 1 and goes to infinity. To find the sum of the series, we can use the formula for the sum of an infinite geometric series. The formula is S = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = 1 and r = -1/6. Plugging these values into the formula, we get S = 1 / (1 + 1/6) = 6/7.
Now, to find the sum of the first four terms, we substitute n = 1, 2, 3, and 4 into the series expression and add them up. Doing the calculations, we get S_4 = (-1)^(1+1)*(1)/(6^(1)) + (-1)^(2+1)*(1)/(6^(2)) + (-1)^(3+1)*(1)/(6^(3)) + (-1)^(4+1)*(1)/(6^(4)) = 1/6 - 1/36 + 1/216 - 1/1296 ≈ 0.157.
The magnitude of the error involved in using the sum of the first four terms to approximate the sum of the entire series is the difference between the two sums, which is |S - S_4|. Substituting the values we found earlier, we get |6/7 - 0.157| ≈ 0.648.