Let's address the problem step by step:

Part (a): Show that the series converges

The given series is:

$\sum_{n=1}^{\infty} \frac{(-1)^n}{8^n n}$

This is an alternating series, as it has the term $(-1)^n$. To prove convergence, we can use the Alternating Series Test (Leibniz's Test). According to this test, an alternating series converges if:

1. The terms $a_n = \frac{1}{8^n n}$ decrease monotonically (i.e., $a_{n+1} < a_n$ for all $n$).
2. $\lim_{n \to \infty} a_n = 0$.

Step 1: Verify $a_n = \frac{1}{8^n n}$ decreases monotonically

For $a_n$ to decrease monotonically, the denominator $8^n n$ must grow faster than the numerator (which is constant, $1$) as $n$ increases. Clearly:

$a_{n+1} = \frac{1}{8^{n+1}(n+1)} < a_n = \frac{1}{8^n n}, \quad \text{because } 8^{n+1} > 8^n \text{ and } (n+1) > n.$

Thus, $a_n$ is monotonically decreasing.

Step 2: Compute $\lim_{n \to \infty} a_n$

$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{8^n n}.$

Since $8^n$ grows exponentially while $n$ grows linearly, the denominator goes to infinity, making the limit $0$.

Thus, both conditions of the Alternating Series Test are satisfied, so the series converges.

Part (b): Find how many terms are needed for the error to be no more than $0.00005$

For an alternating series, the error $R_n$ after $n$ terms is bounded by the first omitted term:

$|R_n| \leq a_{n+1}.$

Here, $a_n = \frac{1}{8^n n}$. We need $a_{n+1} \leq 0.00005$, or:

$\frac{1}{8^{n+1}(n+1)} \leq 0.00005.$

Rearranging:

$8^{n+1}(n+1) \geq \frac{1}{0.00005} = 20000.$

Let’s solve this inequality numerically to find $n$:

1. Start testing $n = 1, 2, 3, \dots$ until the inequality is satisfied.
2. For $n = 6$: $8^{6+1}(6+1) = 8^7 \cdot 7 = 2097152 \cdot 7 = 14680064, \text{ which is greater than 20000.}$

Thus, at least 6 terms are needed to ensure the error is no more than $0.00005$.

Would you like detailed numerical verification or further explanation of any step?

Related Questions:

1. What are other convergence tests that could verify this series' convergence?
2. How can we compute the sum of the first $n$ terms of this series explicitly?
3. What happens to the error if we include one additional term?
4. How does the alternating series test compare to the ratio test in this scenario?
5. What practical applications could such alternating series have in mathematics or physics?

Tip: For numerical problems involving series, always double-check your calculations for higher precision, especially when setting error bounds.