Let's break the problem into two parts as given:

Part (a): Calculating the First Six Terms of the Sequence of Partial Sums

The series given is:

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

Each term of the series is:

$a_n = \frac{(-1)^{n-1}}{n!}$

The partial sums are calculated by summing the first $n$ terms of the series. Let's calculate the first six partial sums:

1. Partial sum $S_1$: $S_1 = a_1 = \frac{(-1)^{1-1}}{1!} = \frac{1}{1} = 1$

2. Partial sum $S_2$: $S_2 = a_1 + a_2 = 1 + \frac{(-1)^{2-1}}{2!} = 1 - \frac{1}{2} = 0.5$

3. Partial sum $S_3$: $S_3 = a_1 + a_2 + a_3 = 1 - \frac{1}{2} + \frac{(-1)^{3-1}}{3!} = 1 - 0.5 + \frac{1}{6} = 0.5 + 0.1667 = 0.6667$

4. Partial sum $S_4$: $S_4 = a_1 + a_2 + a_3 + a_4 = 1 - \frac{1}{2} + \frac{1}{6} + \frac{(-1)^{4-1}}{4!}$ $S_4 = 1 - 0.5 + 0.1667 - 0.0417 = 0.625$

5. Partial sum $S_5$: $S_5 = S_4 + \frac{(-1)^{5-1}}{5!} = 0.625 + \frac{1}{120} = 0.625 + 0.0083 = 0.6333$

6. Partial sum $S_6$: $S_6 = S_5 + \frac{(-1)^{6-1}}{6!} = 0.6333 - \frac{1}{720} = 0.6333 - 0.0014 = 0.6319$

Thus, the first six partial sums are approximately: $1, 0.5, 0.6667, 0.625, 0.6333, 0.6319$

Part (b): Convergence or Divergence

The series alternates in sign and its terms involve factorials in the denominator. Factorials grow extremely fast, making the terms of the series decrease rapidly in magnitude.

1. Since the terms approach zero and alternate in sign, this is an alternating series. By the Alternating Series Test (Leibniz's test), the series converges if:
   • The terms $\frac{1}{n!}$ decrease in magnitude.
   • $\lim_{n \to \infty} a_n = 0$.

Both conditions are satisfied here because: $\lim_{n \to \infty} \frac{1}{n!} = 0$

2. Additionally, this series resembles the alternating expansion of $e^{-1}$ (the exponential function).

Hence, the series converges, and the sum approaches $e^{-1}$ as $n$ increases.

Summary

• Part (a): First six partial sums: $1, 0.5, 0.6667, 0.625, 0.6333, 0.6319$.
• Part (b): The series is convergent, and it converges to $e^{-1}$ ($\approx 0.3679$).

Do you want a further breakdown of the calculations or any additional clarification?

Related Questions:

1. Can you calculate the seventh and eighth partial sums?
2. How does the series relate to $e^{-1}$ explicitly?
3. What are other examples of convergent alternating series?
4. What is the rate of convergence for this series?
5. Could we use a different test for convergence here?

Tip: When verifying convergence of an alternating series, always check if the terms decrease and approach zero!