To solve this problem, we’ll analyze the alternating series remainder to determine how many terms of the given series must be summed to ensure that the remainder is less than $10^{-6}$ in magnitude.

Step 1: Understanding the Series and Alternating Series Remainder Theorem

The series is given by: $\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+3)^4}.$

Since it is an alternating series with terms of the form $\frac{(-1)^k}{(2k+3)^4}$, we can use the Alternating Series Remainder Theorem. This theorem states that for a convergent alternating series, the remainder $R_n$ after summing $n$ terms satisfies: $|R_n| \leq |a_{n+1}|,$ where $a_{n+1}$ is the $(n+1)$-th term of the series. Therefore, to ensure that the remainder is less than $10^{-6}$, we need: $\left|\frac{1}{(2(n+1)+3)^4}\right| < 10^{-6}.$

Step 2: Setting Up the Inequality

The general term of the series is $\frac{1}{(2k+3)^4}$. For the $(n+1)$-th term, this becomes: $\frac{1}{(2(n+1) + 3)^4}.$ We need this term to be less than $10^{-6}$: $\frac{1}{(2(n+1) + 3)^4} < 10^{-6}.$

Step 3: Solving for $n$

1. Invert both sides of the inequality: $(2(n+1) + 3)^4 > 10^6.$

2. Take the fourth root of both sides: $2(n+1) + 3 > 100.$

3. Solve for $n$: $2(n+1) > 97,$ $n+1 > 48.5,$ $n > 47.5.$

Since $n$ must be an integer, we round up to ensure the remainder is below the required tolerance. Thus, we find: $n = 48.$

Conclusion

To ensure the remainder is less than $10^{-6}$ in magnitude, we must sum the first 48 terms of the series.

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Would you like further details or have any questions?

Here are 5 related questions to explore further:

1. How does the Alternating Series Remainder Theorem apply to non-alternating series?
2. Can we estimate the convergence speed for this type of series?
3. What is the practical application of ensuring the remainder is within a specific tolerance?
4. How does this series compare to other convergent series with similar forms?
5. What changes in the solution if the tolerance were different, e.g., $10^{-5}$?

Tip: For convergent alternating series, always check if the absolute value of each term decreases as it progresses, as this confirms the use of the Alternating Series Remainder Theorem.