Develop a series for sinhx using Maclaurin's series. f(x) =sinh x f(0)=sinh 0=(e^0-

step 1 Okay, the question we have to develop a series of our sine hyperbolic x using the McLaurin serie

Develop a series for sinhx using Maclaurin's series. ...

Develop a series for $\sinh x$ using Maclaurin's series. $$ \begin{array}{rl} f(x) & =\sinh x & f(0)=\sinh 0=\frac{e^{0}-e^{-0}}{2}=0 \\ f^{\prime}(x) & =\cosh x & f^{\prime}(0)=\cosh 0=\frac{e^{0}+e^{-0}}{2}=1 \\ f^{\prime \prime}(x) & =\sinh x & f^{\prime \prime}(0)=\sinh 0=0 \\ f^{\prime \prime \prime}(x) & =\cosh x & f^{\prime \prime \prime}(0)=\cosh 0=1 \\ f^{\text {iv }}(x) & =\sinh x & f^{\text {iv }}(0)=\sinh 0=0 \\ f^{v}(x) & =\cosh x & f^{v}(0)=\cosh 0=1 \end{array} $$ Substituting in equation (5) gives: $$ \begin{gathered} \sinh x=f(0)+x f^{\prime}(0)+\frac{x^{2}}{2 !} f^{\prime \prime}(0)+\frac{x^{3}}{3 !} f^{\prime \prime \prime}(0) \\ +\frac{x^{4}}{4 !} f^{\mathrm{iv}}(0)+\frac{x^{5}}{5 !} f^{v}(0)+\cdots \\ =0+(x)(1)+\frac{x^{2}}{2 !}(0)+\frac{x^{3}}{3 !}(1)+\frac{x^{4}}{4 !}(0) \\ +\frac{x^{5}}{5 !}(1)+\cdots \end{gathered} $$ i.e. $\sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\cdots$ (as shown in Chapter 16 )

Develop a series for $\sinh x$ using Maclaurin's series.
$$
\begin{array}{rl}
f(x) & =\sinh x & f(0)=\sinh 0=\frac{e^{0}-e^{-0}}{2}=0 \\
f^{\prime}(x) & =\cosh x & f^{\prime}(0)=\cosh 0=\frac{e^{0}+e^{-0}}{2}=1 \\
f^{\prime \prime}(x) & =\sinh x & f^{\prime \prime}(0)=\sinh 0=0 \\
f^{\prime \prime \prime}(x) & =\cosh x & f^{\prime \prime \prime}(0)=\cosh 0=1 \\
f^{\text {iv }}(x) & =\sinh x & f^{\text {iv }}(0)=\sinh 0=0 \\
f^{v}(x) & =\cosh x & f^{v}(0)=\cosh 0=1
\end{array}
$$
Substituting in equation (5) gives:
$$
\begin{gathered}
\sinh x=f(0)+x f^{\prime}(0)+\frac{x^{2}}{2 !} f^{\prime \prime}(0)+\frac{x^{3}}{3 !} f^{\prime \prime \prime}(0) \\
+\frac{x^{4}}{4 !} f^{\mathrm{iv}}(0)+\frac{x^{5}}{5 !} f^{v}(0)+\cdots \\
=0+(x)(1)+\frac{x^{2}}{2 !}(0)+\frac{x^{3}}{3 !}(1)+\frac{x^{4}}{4 !}(0) \\
+\frac{x^{5}}{5 !}(1)+\cdots
\end{gathered}
$$
i.e. $\sinh x=x+\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}+\cdots$
(as shown in Chapter 16 )

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as sinh x and cosh x, are analogues of the ordinary trigonometric functions but are defined using exponential functions. They have unique properties and come with their own set of identities and series expansions, making them useful in various areas of mathematics and physics.

Derivative Evaluation and Pattern Recognition

When expanding functions into series, evaluating the derivatives at a specific point (in this case, x = 0) is a key step. Recognizing the pattern in these derivatives, such as cyclic repetition or the disappearance of even or odd terms, often simplifies the series. For sinh x, the pattern shows that only the odd-powered terms remain because the even derivatives evaluate to zero at 0.

Maclaurin Series

The Maclaurin series is a special case of the Taylor series where the expansion is about x = 0. It expresses a function as an infinite sum of derivatives evaluated at 0, multiplied by powers of x divided by the corresponding factorial. This series is widely used to represent and approximate functions in a power series form.

Taylor Series

The Taylor series is a method for approximating functions using an infinite sum of terms calculated from the values of the function’s derivatives at a single point. When this point is 0, it becomes the Maclaurin series. This approach allows complex functions to be expressed as sums of polynomials, which simplifies analysis and computation.

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