The series you provided is:

$\sum_{n=1}^{\infty} (-1)^{n-1} \frac{n}{6^n}$

Steps:

1. Graph the Sequence of Terms and Partial Sums:

   • Plot the sequence of terms $a_n = (-1)^{n-1} \frac{n}{6^n}$.
   • Plot the sequence of partial sums $S_n = \sum_{k=1}^{n} (-1)^{k-1} \frac{k}{6^k}$.

2. Estimate the Sum Using the Graph:

   • Use the graph of the partial sums to make a rough estimate of the sum of the series as $n$ increases.

3. Use the Alternating Series Estimation Theorem:

   • Apply the Alternating Series Estimation Theorem to estimate the sum to four decimal places.

Let's start by computing and graphing the sequences.