# Series, Limits, and Residues

Sum[f,{i,i_{min},i_{max}}] | the sum |

Sum[f,{i,i_{min},i_{max},di}] | the sum with i increasing in steps of di |

Sum[f,{i,i_{min},i_{max}},{j,j_{min},j_{max}}] | the nested sum |

Product[f,{i,i_{min},i_{max}}] | the product |

_{min},i

_{max}},{j,j

_{min},j

_{max}}] represents a sum over i and j, which would be written in standard mathematical notation as . Notice that in Wolfram Language notation, as in standard mathematical notation, the range of the

*outermost*variable is given

*first*.

*iterator notation*that the Wolfram Language uses. You will see this notation again when we discuss generating tables and lists using Table ("Making Tables of Values"), and when we describe Do loops ("Repetitive Operations").

{i_{max}} | iterate i _{max} times, without incrementing any variables |

{i,i_{max}} | i goes from 1 to i _{max} in steps of 1 |

{i,i_{min},i_{max}} | i goes from i _{min} to i_{max} in steps of 1 |

{i,i_{min},i_{max},di} | i goes from i _{min} to i_{max} in steps of di |

{i,i_{min},i_{max}},{j,j_{min},j_{max}},… | i goes from i _{min} to i_{max}, and for each such value, j goes from j_{min} to j_{max}, etc. |

*exact*. Given precise input, their results are exact formulas.

*approximate*formula that is valid, say, when the quantity x is small.

Series[expr,{x,x_{0},n}] | find the power series expansion of expr about the point x=x _{0} to at most n ^{th} order |

Normal[series] | truncate a power series to give an ordinary expression |

Series[expr,{x,x_{0},n}] | find the power series expansion of expr about the point x=x _{0} to order at most (x-x_{0})^{n} |

Series[expr,{x,x_{0},n_{x}},{y,y_{0},n_{y}}] | |

find series expansions with respect to y, then x |

^{th}derivative is . Whenever this formula applies, it gives the same results as Series. (For common functions, Series nevertheless internally uses somewhat more efficient algorithms.)

*functions*, which you can, for example, compose or invert.

ComposeSeries[series_{1},series_{2},…] | compose power series |

InverseSeries[series,x] | invert a power series |

*reversion*of power series.

Normal[expr] | convert a power series to a normal expression |

SeriesCoefficient[series,n] | give the coefficient of the n ^{th} order term in a power series |

LogicalExpand[series_{1}==series_{2}] | give the equations obtained by equating corresponding coefficients in the power series |

Solve[series_{1}==series_{2},{a_{1},a_{2},…}] | solve for coefficients in power series |

Sum[expr,{n,n_{min},n_{max}}] | find the sum of expr as n goes from n _{min} to n_{max} |

^{th}term in a sequence as a[n], you can use a

*recurrence equation*to specify how it is related to other terms in the sequence.

RSolve[eqn,a[n],n] | solve a recurrence equation |

*difference equations*in which the arguments of differ by integers, but also

*‐difference equations*in which the arguments of are related by multiplicative factors.

RSolve[{eqn_{1},eqn_{2},…},{a_{1}[n],a_{2}[n],…},n] | |

solve a coupled system of recurrence equations |

RSolve[eqns,a[n_{1},n_{2},…],{n_{1},n_{2},…}] | |

solve partial recurrence equations |

*limit*.

Limit[expr,x->x_{0}] | find the limit of expr when x approaches x _{0} |

_{min},x

_{max}}] represents an uncertain value which lies somewhere in the interval to .

Limit[expr,x->x_{0},Direction->1] | find the limit as x approaches x _{0} from below |

Limit[expr,x->x_{0},Direction->-1] | find the limit as x approaches x _{0} from above |

_{0}] tells you what the value of expr is when x tends to x

_{0}. When this value is infinite, it is often useful instead to know the

*residue*of expr when x equals x

_{0}. The residue is given by the coefficient of in the power series expansion of expr about the point x

_{0}.

Residue[expr,{x,x_{0}}] | the residue of expr when x equals x _{0} |

PadeApproximant[f,{x,x_{0},{n,m}}] | give the Padé approximation to centered at x _{0} of order (n,m) |

PadeApproximant[f,{x,x_{0},n}] | give the diagonal Padé approximation to centered at x _{0} of order n |