The problem in the image involves the infinite series:

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

Solution:

This is an alternating series, and the general term $a_n$ is given by:

$a_n = \frac{n}{6^n}$

1. Check for Convergence:

To determine if the series converges, we can apply the Alternating Series Test (Leibniz criterion). The series $\sum (-1)^{n-1} a_n$ will converge if:

• The sequence $a_n = \frac{n}{6^n}$ is decreasing.
• $\lim_{n \to \infty} a_n = 0$.

Let's check these conditions:

1. Monotonic Decreasing:

   • We need to check if $a_{n+1} < a_n$ for all $n$. $a_{n+1} = \frac{n+1}{6^{n+1}}, \quad a_n = \frac{n}{6^n}$ We compare the ratio: $\frac{a_{n+1}}{a_n} = \frac{\frac{n+1}{6^{n+1}}}{\frac{n}{6^n}} = \frac{n+1}{6n}$ Since $\frac{n+1}{6n} < 1$ for all $n$, $a_n$ is indeed decreasing.

2. Limit to Zero: $\lim_{n \to \infty} \frac{n}{6^n} = 0$ As $n$ grows, the exponential $6^n$ grows much faster than $n$, so the limit is zero.

Both conditions are satisfied, so the series converges.

2. Estimate the Sum using the Alternating Series Estimation Theorem:

The Alternating Series Estimation Theorem states that the error $R_n$ when approximating the sum $S$ of the infinite series by the $n$-th partial sum $S_n$ is less than the magnitude of the first omitted term $a_{n+1}$.

So, if we want to estimate the sum up to $n$ terms, the error is less than $a_{n+1} = \frac{n+1}{6^{n+1}}$.

3. Graphing the Sequence and the Partial Sums:

The problem asks to graph the sequence of terms $a_n = \frac{n}{6^n}$ and the partial sums $S_n$. These graphs would typically show:

• The sequence $a_n$ decreasing rapidly towards zero.
• The partial sums $S_n$ converging to the actual sum of the series.

To perform these calculations and graph the results, you'd typically use a software tool or graphing calculator. You could also plot the first few terms manually.

Final Answer:

• Convergence: The series converges.
• Sum Estimation: Use the Alternating Series Estimation Theorem to approximate the sum $S$ by calculating partial sums and ensuring the error $R_n$ is less than $a_{n+1}$.

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Would you like to see the partial sums calculated for specific terms or need more details on the estimation process?

Here are some related questions:

1. What is the Alternating Series Test?
2. How do you determine if a series is monotonic?
3. Why does $\lim_{n \to \infty} \frac{n}{6^n} = 0$?
4. How does the Alternating Series Estimation Theorem work?
5. What is the significance of the first omitted term in series estimation?
6. How do you plot partial sums for a series?
7. What is the difference between absolute and conditional convergence?
8. How do exponential functions compare to polynomial growth?

Tip: When estimating the sum of an alternating series, remember that the error is always less than the first omitted term, which can greatly simplify calculations!