Use the Maclaurin series for 

*ex*

 to calculate 

1/

10

*e*

 correct to five decimal places. (Round your answer to five decimal places.)

To calculate \(\frac{1}{10e}\) using the Maclaurin series for \(e^x\) correct to five decimal places, we'll follow these steps:

### Step 1: Write the Maclaurin series for \(e^x\)

The Maclaurin series for \(e^x\) is:

\[
e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots
\]

### Step 2: Evaluate the series for \(e^{1/10}\)

Set \(x = \frac{1}{10} = 0.1\), and substitute it into the series:

\[
e^{0.1} = 1 + 0.1 + \frac{(0.1)^2}{2!} + \frac{(0.1)^3}{3!} + \frac{(0.1)^4}{4!} + \cdots
\]

Let's calculate the first few terms:

\[
1 + 0.1 + \frac{0.01}{2} + \frac{0.001}{6} + \frac{0.0001}{24} + \cdots
\]

This simplifies to:

\[
e^{0.1} \approx 1 + 0.1 + 0.005 + 0.0001667 + 0.00000417
\]

\[
e^{0.1} \approx 1.10517083
\]

### Step 3: Calculate \(\frac{1}{10e}\)

Now, to find \(\frac{1}{10e}\), we calculate:

\[
\frac{1}{10e} = \frac{1}{10 \times e^{0.1}} = \frac{1}{1.10517083}
\]

Using a calculator or by performing the division:

\[
\frac{1}{1.10517083} \approx 0.90484
\]

### Final Answer:

The value of \(\frac{1}{10e}\) correct to five decimal places is approximately:

\[
\boxed{0.09048}
\]

Would you like further details or have any questions?

---

Here are some related questions you might be interested in:

1. How does the accuracy of the Maclaurin series change with more terms?
2. Can you find \(\frac{1}{e}\) using the same method?
3. How is the Maclaurin series related to the Taylor series?
4. What is the significance of the error term in the Maclaurin series approximation?
5. How can you use the Maclaurin series to approximate other exponential functions?
6. What is the radius of convergence for the Maclaurin series of \(e^x\)?
7. Can you derive the Maclaurin series for \(\cos(x)\) or \(\sin(x)\)?
8. What are the applications of the exponential function in real-life scenarios?

**Tip:** When using a series expansion to approximate values, it's important to consider the remainder term to understand how many terms are necessary for a desired accuracy.

Learn how to calculate 1/(10e) using the Maclaurin series for ex correct to five decimal places. Follow step-by-step instructions with detailed mathematical formulas and series approximation techniques. Suitable for advanced high school students and those interested in mathematical series expansions.

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