Using   the   part   (i),    write   down   the   Maclaurin   series   of   cosh(x), where  cosh(x)=$$ \frac12 $$($$ e^{x} $$ + $$ e^{-x} $$)

To find the Maclaurin series for $\cosh(x)$, we can use the given definition:

\[
\cosh(x) = \frac{1}{2} \left( e^x + e^{-x} \right)
\]

We already know the Maclaurin series for $e^x$:

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

The Maclaurin series for $e^{-x}$ can be obtained by replacing $x$ with $-x$ in the series for $e^x$:

\[
e^{-x} = 1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} - \cdots
\]

Now, add the series for $e^x$ and $e^{-x}$:

\[
e^x + e^{-x} = \left(1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots\right) + \left(1 - \frac{x}{1!} + \frac{x^2}{2!} - \frac{x^3}{3!} + \frac{x^4}{4!} - \cdots\right)
\]

This simplifies to:

\[
e^x + e^{-x} = 2 + \frac{2x^2}{2!} + \frac{2x^4}{4!} + \cdots
\]

Finally, divide by 2 to get $\cosh(x)$:

\[
\cosh(x) = \frac{1}{2} \left(e^x + e^{-x}\right) = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots
\]

Thus, the Maclaurin series for $\cosh(x)$ is:

\[
\cosh(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!}
\]

### Do you want more details or have any questions?

Here are some related questions:

1. What is the Maclaurin series for $\sinh(x)$?
2. How can you use the Maclaurin series to approximate $\cosh(x)$ for small values of $x$?
3. What are the derivatives of $\cosh(x)$ at $x = 0$?
4. How does the series for $\cosh(x)$ compare to the series for $\cos(x)$?
5. What is the radius of convergence for the Maclaurin series of $\cosh(x)$?

**Tip:** For even functions like $\cosh(x)$, only even powers of $x$ appear in the series expansion, making it easier to compute specific values.

Understand the Maclaurin series of cosh(x) using its definition and series expansions for exponential functions. This detailed solution covers the derivation of the Maclaurin series for hyperbolic cosine, essential for advanced mathematics students and enthusiasts.

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