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To solve this problem, we need to find the Taylor series expansion of the function \( f(z) = \frac{e^z}{1-z} \) about \( z = 0 \).

### Step 1: Express the function as a product of two known series
We can write the function \( f(z) \) as the product of two series:
\[
f(z) = \frac{e^z}{1-z} = e^z \cdot \frac{1}{1-z}
\]

### Step 2: Find the Taylor series expansion of \( e^z \)
The Taylor series expansion of \( e^z \) around \( z = 0 \) is:
\[
e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!}
\]

### Step 3: Find the Taylor series expansion of \( \frac{1}{1-z} \)
The Taylor series expansion of \( \frac{1}{1-z} \) around \( z = 0 \) is a geometric series:
\[
\frac{1}{1-z} = \sum_{n=0}^{\infty} z^n
\]

### Step 4: Multiply the two series
Now, we multiply the two series:
\[
f(z) = \left( \sum_{n=0}^{\infty} \frac{z^n}{n!} \right) \left( \sum_{m=0}^{\infty} z^m \right)
\]
\[
f(z) = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{z^{n+m}}{n!}
\]

### Step 5: Combine terms of the same power of \( z \)
Combine the terms where the power of \( z \) is the same:
\[
f(z) = \sum_{k=0}^{\infty} \left( \sum_{n=0}^{k} \frac{1}{n!} \right) z^k
\]

So, the Taylor series for \( f(z) = \frac{e^z}{1-z} \) about \( z = 0 \) is:
\[
f(z) = \sum_{k=0}^{\infty} \left( \sum_{n=0}^{k} \frac{1}{n!} \right) z^k
\]

This is the required Taylor series expansion for the given function.

Do you want more details or have any questions?

Here are 5 related questions for further practice:
1. Find the Taylor series expansion of \( \frac{e^z}{1+z} \) about \( z = 0 \).
2. Determine the radius of convergence for the series expansion of \( f(z) = \frac{e^z}{1-z} \).
3. Expand \( \sin(z) \cdot \frac{1}{1+z} \) as a Taylor series around \( z = 0 \).
4. Compute the first four non-zero terms of the Taylor series of \( \frac{e^{2z}}{1-z} \) about \( z = 0 \).
5. Explain how to derive the Taylor series for a product of two functions.

**Tip:** Always check the radius of convergence when dealing with series expansions to ensure the series is valid in the desired interval.

Explore the Taylor series expansion of the function f(z) = e^z / (1-z) around z = 0. This detailed solution includes step-by-step calculations and explanations of Taylor series concepts, exponential functions, and geometric series. Perfect for advanced mathematics students and enthusiasts.

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