The Taylor series expansion is the most common way to approximate
an expression by a polynomial. However, not all expressions can be
represented by Taylor series. For example, you cannot compute a Taylor
series expansion for the following expression around ```
x =
2
```

:

taylor(1/(x^3 - 8), x = 2)

Error: Cannot compute a Taylor expansion of '1/(x^3 - 8)'. Try 'series' for a more general expansion. [taylor]

If a Taylor series expansion does not exist for your expression,
try to compute other power series. MuPAD^{®} provides the function `series`

for computing
power series. When you call `series`

, MuPAD tries
to compute the following power series:

Taylor series

Laurent series

Puiseux series. For more information, see

`Series::Puiseux`

.Generalized series expansion of

`f`

around*x*=*x*_{0}. For more information, see`Series::gseries`

.

As soon as `series`

computes
any type of power series, it does not continue computing other types
of series, but stops and returns the result. For example, for this
expression it returns a Laurent series:

S := series(1/(x^3 - 8), x = 2); testtype(S, Type::Series(Laurent))

When computing series expansions, MuPAD returns only those
results that are valid for all complex values of the expansion variable
in some neighborhood of the expansion point. If you need the expansion
to be valid only for real numbers, use the option `Real`

.
For example, when you compute the series expansion of the following
expression for complex numbers, `series`

returns:

series(sign(x^2*sin(x)), x = 0)

When you compute the series expansion for real numbers, `series`

returns a simplified
result:

series(sign(x^2*sin(x)), x = 0, Real)

Along the real axis, compute series expansions for this expression
when `x`

approaches the value 0 from the left and
from the right sides:

series(sign(x^2*sin(x)), x = 0, Left); series(sign(x^2*sin(x)), x = 0, Right)

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