To expand \(\sin^{-1}(z)\) (the inverse sine function, also denoted as \(\arcsin(z)\)) in terms of a series, we use the Taylor series expansion around \(z = 0\). The series expansion for \(\sin^{-1}(z)\) is given by:

\[
\sin^{-1}(z) = \sum_{n=0}^{\infty} \frac{(2n)!}{4^n (n!)^2 (2n+1)} z^{2n+1}
\]

Let's break down the expansion:

- **\(n = 0\):** 
  \[
  \frac{(2 \cdot 0)!}{4^0 (0!)^2 (2 \cdot 0 + 1)} z^{2 \cdot 0 + 1} = z
  \]

- **\(n = 1\):** 
  \[
  \frac{(2 \cdot 1)!}{4^1 (1!)^2 (2 \cdot 1 + 1)} z^{2 \cdot 1 + 1} = \frac{1}{6} z^3
  \]

- **\(n = 2\):** 
  \[
  \frac{(2 \cdot 2)!}{4^2 (2!)^2 (2 \cdot 2 + 1)} z^{2 \cdot 2 + 1} = \frac{3}{40} z^5
  \]

- **\(n = 3\):** 
  \[
  \frac{(2 \cdot 3)!}{4^3 (3!)^2 (2 \cdot 3 + 1)} z^{2 \cdot 3 + 1} = \frac{5}{112} z^7
  \]

Putting it all together, we have:

\[
\sin^{-1}(z) = z + \frac{1}{6}z^3 + \frac{3}{40}z^5 + \frac{5}{112}z^7 + \cdots
\]

This series converges for \(|z| \leq 1\), but is most practical for \(|z| < 1\) to ensure rapid convergence.

Would you like further details or have any specific questions?

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**Related Questions:**

1. What is the radius of convergence for the series expansion of \(\sin^{-1}(z)\)?
2. How do you derive the general term in the series expansion for \(\sin^{-1}(z)\)?
3. Can \(\sin^{-1}(z)\) be expanded using complex analysis?
4. What is the series expansion for \(\cos^{-1}(z)\)?
5. How does the series expansion of \(\sin^{-1}(z)\) compare with other inverse trigonometric functions?
6. What are the practical applications of the series expansion of \(\sin^{-1}(z)\)?
7. How can we use the series expansion of \(\sin^{-1}(z)\) to approximate angles?
8. What is the relationship between the series expansion of \(\sin^{-1}(z)\) and power series?

**Tip:** When working with series expansions, it's essential to consider the range of convergence to ensure the validity of the expansion in your calculations.

Learn how to expand sin^-1(z) (arcsin(z)) in terms of z using the Taylor series expansion. This detailed solution covers the series expansion formula and its application, suitable for advanced undergraduate level mathematics.

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