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# Documentation

## Taylor Series

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):

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

Technically, T is a Maclaurin series, since its expansion point is a = 0.

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);
ezplot(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.