Limits of Sequences¶
Provides methods to compute limit of terms having sequences at infinity.
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sympy.series.limitseq.
difference_delta
(expr, n=None, step=1)[source]¶ Difference Operator.
Discrete analogous to differential operator.
References
[R433] https://reference.wolfram.com/language/ref/DifferenceDelta.html Examples
>>> from sympy import difference_delta as dd >>> from sympy.abc import n >>> dd(n*(n + 1), n) 2*n + 2 >>> dd(n*(n + 1), n, 2) 4*n + 6
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sympy.series.limitseq.
dominant
(expr, n)[source]¶ Finds the most dominating term in an expression.
if limit(a/b, n, oo) is oo then a dominates b. if limit(a/b, n, oo) is 0 then b dominates a. else a and b are comparable.
returns the most dominant term. If no unique domiant term, then returns
None
.See also
Examples
>>> from sympy import Sum >>> from sympy.series.limitseq import dominant >>> from sympy.abc import n, k >>> dominant(5*n**3 + 4*n**2 + n + 1, n) 5*n**3 >>> dominant(2**n + Sum(k, (k, 0, n)), n) 2**n
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sympy.series.limitseq.
limit_seq
(expr, n=None, trials=5)[source]¶ Finds limits of terms having sequences at infinity.
Parameters: expr : Expr
SymPy expression that is admissible (see section below).
n : Symbol
Find the limit wrt to n at infinity.
trials: int, optional
The algorithm is highly recursive.
trials
is a safeguard from infinite recursion incase limit is not easily computed by the algorithm. Try increasingtrials
if the algorithm returnsNone
.See also
References
[R434] Computing Limits of Sequences - Manuel Kauers Examples
>>> from sympy import limit_seq, Sum, binomial >>> from sympy.abc import n, k, m >>> limit_seq((5*n**3 + 3*n**2 + 4) / (3*n**3 + 4*n - 5), n) 5/3 >>> limit_seq(binomial(2*n, n) / Sum(binomial(2*k, k), (k, 1, n)), n) 3/4 >>> limit_seq(Sum(k**2 * Sum(2**m/m, (m, 1, k)), (k, 1, n)) / (2**n*n), n) 4
Admissible Terms
The terms should be built from rational functions, indefinite sums, and indefinite products over an indeterminate n. A term is admissible if the scope of all product quantifiers are asymptotically positive. Every admissible term is asymptoticically monotonous.