Question

$\mathbf{cosh}\mathit{x}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{x}}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{x}}}{\mathbf{2}}$

Short answer

The Maclaurin series, i.e., $\frac{e^x+e^{-x}}{2}=1+\frac{x^2}{2}+\frac{x^3}{24}+\frac{x^6}{720}+\dots$.

The general term, i.e., $\frac{e^x+e^{-x}}{2}=\sum _{n=0}^{\infty }\frac{x^{2n}}{\left(2n\right)!}$

The Maclaurin series using MATLAB tool, i.e.,

Graph of the function and $s_1$ together, i.e.,

Graph of the function and $s_2$ together, i.e.,

Graph of the function and $s_3$ together, i.e.,

Step-by-step solution

Given Information.

The given function.

$\cosh x=\frac{e^x+e^{-x}}{2}$

Definition of Maclaurin series.

All of the terms in a Maclaurin series are nonnegative integer powers of the variable. It's a Taylor series expansion of a function with a value of around 0.

Find the Maclaurin series for coshx=ex+e-x2 .

Use the Maclaurin series of $e^x$ and replace $x$ by$-x$ ,

$f\left(x\right)=f\left(0\right)+\frac{f^{\text{'}}\left(0\right)}{1!}\left(x\right)+\frac{f^{\text{'}\text{'}}\left(0\right)}{2!}\left(x^2\right)+\frac{f^{\text{'}\text{'}\text{'}}\left(0\right)}{3!}\left(x^3\right)+\dots$

localid="1657469142014" $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$

$e^{-x}=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}+\dots$

Further, add the above expressions.

$e^x+e^{-x}=\left[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\dots \right]+\left[1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}+\dots \right]$

$e^x+e^{-x}=2+\frac{2x^2}{2!}+\frac{2x^4}{4!}+\dots$

$\frac{e^x+e^{-x}}{2}=1+\frac{x^2}{2}+\frac{x^3}{24}+\frac{x^6}{720}+\dots$

Write the general term of coshx=ex+e-x2   .

Use the general term of $e^x$ and replace $x$ by $-x$.

role="math" localid="1657469609662" $e^x=\sum _{n=0}^{\infty }\frac{x^n}{\left(n\right)!}$

role="math" localid="1657470247728" $$\begin{gathered} \cos h\left(x\right)=\frac{e^x+e^{-x}}{2} \\ =\sum _{n=0}^{\infty }\frac{x^{2n}}{\left(2n\right)!} \end{gathered}$$

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Use MATLAB tool to find Maclaurin series.

Find first few terms of Maclaurin series.

Approximate partial sums of the series.

Write partial sums of the series$[s_1,s_2,s_3]$ .

$$\begin{gathered} s_1=1+\frac{x^2}{2} \\ {s}_2=1+\frac{x^2}{2}+\frac{x^3}{24} \\ {s}_3=1+\frac{x^2}{2}+\frac{x^3}{24}+\frac{x^6}{720}+\dots \end{gathered}$$

Plot the function coshx=ex+e-x2  and s1 together.

Graph of the function and $s_1$ together are shown as:

Plot the function coshx=ex+e-x2  and s2 together.

Graph of the function and $s_2$ together are shown as:

Plot the function coshx=ex+e-x2  and s3 together.

Graph of the function and $s_3$ together are shown as: