Let's solve the problem step by step.

Problem Statement Recap

We are given the following alternating series:

$\sum_{k=0}^{\infty} \frac{(-1)^k}{(4k + 2)^4}$

We need to:

1. Find the 2nd partial sum $S_2$.
2. Determine an upper bound for the error $|S - S_n|$ when approximating the series $S$ with the $n$-th partial sum $S_n$, for $n = 2$.

Step 1: Finding the 2nd Partial Sum $S_2$

The nth partial sum $S_n$ for an alternating series is the sum of the first $n$ terms of the series.

For $n = 2$, we calculate the partial sum $S_2$:

$S_2 = \frac{(-1)^0}{(4(0) + 2)^4} + \frac{(-1)^1}{(4(1) + 2)^4}$

This simplifies to:

$S_2 = \frac{1}{2^4} - \frac{1}{6^4}$

Let's calculate this:

$\frac{1}{2^4} = \frac{1}{16} = 0.0625$ $\frac{1}{6^4} = \frac{1}{1296} \approx 0.0007716$

So:

$S_2 = 0.0625 - 0.0007716 \approx 0.0617284$

Step 2: Finding the Upper Bound for the Error

The error bound for an alternating series is given by the absolute value of the first omitted term:

$|S - S_2| \leq \left| \frac{(-1)^2}{(4(2) + 2)^4} \right|$

Simplifying the expression inside the absolute value:

$\left| \frac{1}{(8 + 2)^4} \right| = \frac{1}{10^4} = \frac{1}{10000} = 0.0001$

Final Answers

• The 2nd partial sum $S_2$ is approximately 0.0617284.
• The upper bound for the error $|S - S_2|$ is 0.0001.

Do you want details or have any questions?

Here are 5 related questions you might want to consider:

1. What happens to the partial sum as $n$ increases?
2. How can the error bound be decreased further?
3. What is the general form of the $n$-th term of this series?
4. How does the error bound affect the accuracy of the partial sum?
5. Can this series be expressed in a closed form?

Tip: When evaluating alternating series, the error bound is a useful tool to determine how accurate your partial sum is.