The statements

syms x f = 1/(5 + 4*cos(x)); T = taylor(f, 'Order', 8)

return

T = (49*x^6)/131220 + (5*x^4)/1458 + (2*x^2)/81 + 1/9

which is all the terms up to, but not including, order eight
in the Taylor series for *f*(*x*):

$$\sum _{n=0}^{\infty}{(x-a)}^{n}\frac{{f}^{(n)}(a)}{n!}}.$$

The command

pretty(T)

prints `T`

in a format resembling typeset mathematics:

6 4 2 49 x 5 x 2 x 1 ------ + ---- + ---- + - 131220 1458 81 9

These commands

syms x g = exp(x*sin(x)); t = taylor(g, 'ExpansionPoint', 2, 'Order', 12);

generate the first 12 nonzero terms of the Taylor series for `g`

about ```
x
= 2
```

.

`t`

is a large expression; enter

size(char(t))

ans = 1 99791

to find that `t`

has about 100,000 characters
in its printed form. In order to proceed with using `t`

,
first simplify its presentation:

t = simplify(t); size(char(t))

ans = 1 6988

Next, plot these functions together to see how well this Taylor
approximation compares to the actual function `g`

:

xd = 1:0.05:3; yd = subs(g,x,xd); fplot(t, [1, 3]) hold on plot(xd, yd, 'r-.') title('Taylor approximation vs. actual function') legend('Taylor','Function')

Special thanks is given to Professor Gunnar Bäckstrøm of UMEA in Sweden for this example.

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