Question

Graph both the sequence of terms and the sequence of partial sums on the same screen. Use the graph to make a rough estimate of the sum of the series. Then use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places. ∑[infinity]ₙ₌₁ ​ (−1)ⁿ−¹ n/6ⁿ

Answer 1

To graph the sequence of terms and partial sums of the given series and estimate its sum using the Alternating Series Estimation Theorem.

Step-by-step explanation:

To graph both the sequence of terms and the sequence of partial sums of the series ∑ₙ₌₁​ (−1)ⁿ⁻¹ n/6ⁿ, we can calculate the terms and partial sums for various values of n.

The terms of the series can be written as follows: (-1)ⁿ⁻¹ n/6ⁿ. For example, when n=1, the first term is (-1)⁰ 1/6ⁱ = 1/6, when n=2, the second term is (-1)¹ 2/6² = -2/36, and so on.

Similarly, the partial sums of the series can be calculated by adding up the terms. For example, for n=1, the first partial sum is 1/6, for n=2, the second partial sum is 1/6 - 2/36, and so on.

Once we have the values for the terms and the partial sums, we can plot them on a graph. We can then estimate the sum of the series by observing the trend of the partial sums. Finally, we can use the Alternating Series Estimation Theorem to estimate the sum correct to four decimal places.