1 | initial version |

Use the `taylor`

method to get a Taylor polynomial. This needs to be done in the symbolic ring since the power series ring is purely algebraic and doesn't know about derivatives.

```
sage: t = var('t')
sage: f = sin(t)
sage: s_poly = f.taylor(t,0,7)
sage: s_poly
-1/5040*t^7 + 1/120*t^5 - 1/6*t^3 + t
sage: s_poly.parent()
Symbolic Ring
```

The `power_series`

method will convert the Taylor polynomial to a power series for you:

```
sage: s_ser = s_poly.power_series(QQ)
sage: s_ser
t - 1/6*t^3 + 1/120*t^5 - 1/5040*t^7 + O(t^8)
sage: s_ser.parent()
Power Series Ring in t over Rational Field
sage: g = cos(t)
sage: c_poly = g.taylor(t,0,7)
sage: c_poly
-1/720*t^6 + 1/24*t^4 - 1/2*t^2 + 1
sage: c_ser = c_poly.power_series(QQ)
sage: c_ser
1 - 1/2*t^2 + 1/24*t^4 - 1/720*t^6 + O(t^7)
```

Over the power series ring, you can do arithmetic with the power series. Note that you will need to change "`t`

" from the generator of the symbolic ring to the generator of the power series ring in order to use things like `t+2`

:

```
sage: t.parent()
Symbolic Ring
sage: t = s_ser.parent().gen()
sage: t.parent()
Power Series Ring in t over Rational Field
sage: s_ser/c_ser(t+2)
-45/19*t + 1890/361*t^2 - 83085/6859*t^3 + 3677670/130321*t^4 - 649441779/9904396*t^5 + 7173329982/47045881*t^6 - 17745663554041/50056817384*t^7 + O(t^8)
```

Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.