--- title: "DiscreteAsymptotic" language: "en" type: "Symbol" summary: "DiscreteAsymptotic[expr, n -> \\[Infinity]] gives an asymptotic approximation for expr as n tends to infinity over the integers. DiscreteAsymptotic[expr, {n, \\[Infinity], m}] gives an asymptotic series approximation for expr to order m." keywords: - asymptotic expansion - series expansion - leading order term - perturbation expansion - asymptotic sum - asymptotic transform - asymptotic solution - discrete series canonical_url: "https://reference.wolfram.com/language/ref/DiscreteAsymptotic.html" source: "Wolfram Language Documentation" related_guides: - title: "Asymptotics" link: "https://reference.wolfram.com/language/guide/Asymptotics.en.md" related_functions: - title: "Asymptotic" link: "https://reference.wolfram.com/language/ref/Asymptotic.en.md" - title: "Series" link: "https://reference.wolfram.com/language/ref/Series.en.md" - title: "DiscreteLimit" link: "https://reference.wolfram.com/language/ref/DiscreteLimit.en.md" - title: "AsymptoticSum" link: "https://reference.wolfram.com/language/ref/AsymptoticSum.en.md" - title: "AsymptoticProduct" link: "https://reference.wolfram.com/language/ref/AsymptoticProduct.en.md" - title: "AsymptoticRSolveValue" link: "https://reference.wolfram.com/language/ref/AsymptoticRSolveValue.en.md" - title: "Sum" link: "https://reference.wolfram.com/language/ref/Sum.en.md" - title: "Product" link: "https://reference.wolfram.com/language/ref/Product.en.md" - title: "RSolveValue" link: "https://reference.wolfram.com/language/ref/RSolveValue.en.md" - title: "SeriesCoefficient" link: "https://reference.wolfram.com/language/ref/SeriesCoefficient.en.md" - title: "InverseZTransform" link: "https://reference.wolfram.com/language/ref/InverseZTransform.en.md" - title: "Prime" link: "https://reference.wolfram.com/language/ref/Prime.en.md" --- # DiscreteAsymptotic DiscreteAsymptotic[expr, n -> ∞] gives an asymptotic approximation for expr as n tends to infinity over the integers. DiscreteAsymptotic[expr, {n, ∞, m}] gives an asymptotic series approximation for expr to order m. ## Details and Options * ``DiscreteAsymptotic`` is typically used to solve problems for which no exact solution can be found or to get simpler answers for computation, comparison and interpretation. In such cases, an asymptotic approximation often gives enough information for simplifying or solving application problems. * ``DiscreteAsymptotic[expr, n -> ∞]`` computes the leading term in an asymptotic expansion for ``expr``. Use ``SeriesTermGoal`` to specify more terms. * The expression ``expr`` can be any sequence $a(n)$, a sum specified by ``Sum``, a product specified by ``Product``, a sequence specified by ``SeriesCoefficient``, a difference equation specified by ``RSolveValue``, etc. * If the exact result is ``g[x]`` and the asymptotic approximation of order ``n`` at ``x0`` is ``gn[x]``, then ``AsymptoticLess[g[x] - gn[x], gn[x] - gn - 1[x], x -> x0]`` or ``g[x] - gn[x]∈o[gn[x] - gn - 1[x]]`` as ``x -> x0``. [image] * The asymptotic approximation ``gn[x]`` is often given as a sum ``gn[x] == ∑k = 1nαkϕk[x]``, where ``{ϕ1[x], …, ϕn[x]}`` is an asymptotic scale ``ϕ1[x]≻ϕ2[x]≻⋯ > ϕn[x]`` as ``x -> x0``. Then ``AsymptoticLess[g[x] - gn[x], ϕn[x], x -> x0]`` or ``g[x] - gn[x]∈o[ϕn[x]]`` as ``x -> x0``. * Common asymptotic scales include: | | | | --- | --- | | $\left(x-x_0\right){}^0\succ \left(x-x_0\right){}^1\succ \left(x-x_0\right){}^2\succ \cdots$ | Taylor scale when x -> x0 | | $\left(x-x_0\right){}^{-3}\succ \left(x-x_0\right){}^{-2}\succ \left(x-x_0\right){}^{-1}\succ \cdots$ | Laurent scale when x -> x0 | | $x^{-1}\succ x^{-2}\succ x^{-3}\succ \cdots$ | Laurent scale when x -> ±∞ | | $\left(x-x_0\right){}^{1/p}\succ \left(x-x_0\right){}^{2/p}\succ \left(x-x_0\right){}^{3/p}\succ \ldots$ | Puiseux scale when x -> x0 | * The scales used to express the asymptotic approximation are automatically inferred from the problem and can often include more exotic scales. * The following options can be given: | | | | | ------------------- | ----------------- | ------------------------------------------------------------------ | | AccuracyGoal | Automatic | digits of absolute accuracy sought | | Assumptions | \$Assumptions | assumptions to make about parameters | | GenerateConditions | Automatic | whether to generate answers that involve conditions on parameters | | GeneratedParameters | None | how to name generated parameters | | Method | Automatic | method to use | | PerformanceGoal | \$PerformanceGoal | aspects of performance to optimize | | PrecisionGoal | Automatic | digits of precision sought | | SeriesTermGoal | Automatic | number of terms in the approximation | | WorkingPrecision | Automatic | the precision used in internal computations | * With the default setting of ``Automatic`` for ``GenerateConditions``, conditions on parameters are typically not returned in the results from ``DiscreteAsymptotic``. Answers that include conditions on parameters may be obtained by setting ``GenerateConditions`` to ``True``. * Possible settings for ``PerformanceGoal`` include ``\$PerformanceGoal``, ``"Quality"`` and ``"Speed"``. With the ``"Quality"`` setting, ``DiscreteAsymptotic`` typically solves more problems or produces simpler results, but it potentially uses more time and memory. * With the default setting of ``Automatic`` for ``WorkingPrecision``, ``AccuracyGoal`` and ``PrecisionGoal``, ``DiscreteAsymptotic`` may return an asymptotic approximation with a lower precision, even if the input has infinite precision. ## Examples (29) ### Basic Examples (4) Find the leading asymptotic term for $n!$ as $n$ approaches ``Infinity`` : ```wl In[1]:= DiscreteAsymptotic[n!, n -> ∞] Out[1]= E^-n n^(1/2) + n Sqrt[2 π] ``` Compare the values of the sequence, the approximation and their ratios: ```wl In[2]:= Table[{n!, N[%], n! / N[%]}, {n, 1, 21, 4}]//TableForm Out[2]//TableForm= | | | | | :------------------- | :--------------------- | :------ | | 1 | 0.922137 | 1.08444 | | 120 | 118.019 | 1.01678 | | 362880 | 359537. | 1.0093 | | 6227020800 | 6.187239475192709*^9 | 1.00643 | | 355687428096000 | 3.5394832866610056*^14 | 1.00491 | | 51090942171709440000 | 5.088861732550965*^19 | 1.00398 | ``` Use ``SeriesTermGoal`` to obtain more terms from the expansion: ```wl In[3]:= DiscreteAsymptotic[n!, n -> ∞, SeriesTermGoal -> 3] Out[3]= E^-n (1 + (1/288 n^2) + (1/12 n)) n^(1/2) + n Sqrt[2 π] ``` --- Find the asymptotic behavior of a sequence using its generating function: ```wl In[1]:= genfun = (1/2 - E^z); ``` Apply ``DiscreteAsymptotic`` to compute an asymptotic approximation: ```wl In[2]:= asy = DiscreteAsymptotic[$$\text{SeriesCoefficient}[\text{genfun},\{z,0,n\}]$$, n -> ∞] Out[2]= (1/2) Log[2]^-1 - n In[3]:= Table[asy, {n, 0, 20, 5}]//N Out[3]= {0.721348, 4.50835, 28.1767, 176.101, 1100.61} ``` Compare the approximation with the values of the sequence: ```wl In[4]:= Table[SeriesCoefficient[genfun, {z, 0, n}], {n, 0, 20, 5}] Out[4]= {1, (541/120), (34082521/1209600), (17714091613681/100590336000), (16228410886329601231/14744860655616000)} In[5]:= N[%] Out[5]= {1., 4.50833, 28.1767, 176.101, 1100.61} ``` --- Compute an asymptotic approximation for a definite sum: ```wl In[1]:= DiscreteAsymptotic[Inactive[Sum][Binomial[n, k] ^ 2, {k, 0, n}], n -> ∞] Out[1]= (4^n/Sqrt[n] Sqrt[π]) ``` Obtain the same result using ``AsymptoticSum`` : ```wl In[2]:= AsymptoticSum[Binomial[n, k] ^ 2, {k, 0, n}, n -> ∞] Out[2]= (4^n/Sqrt[n] Sqrt[π]) ``` --- Compute an asymptotic approximation for a difference equation: ```wl In[1]:= dr = DifferenceRoot[Function[{y, n}, {y[n + 1] - n ^ 2 y[n] == 0, y[1] == 1}]][n]; In[2]:= DiscreteAsymptotic[dr, n -> ∞] Out[2]= 2 E^-2 n n^-1 + 2 n π ``` Obtain the same result using ``AsymptoticRSolveValue`` : ```wl In[3]:= AsymptoticRSolveValue[{y[n + 1] - n ^ 2 * y[n] == 0, y[1] == 1}, y[n], n -> ∞] Out[3]= 2 E^-2 n n^-1 + 2 n π ``` ### Scope (19) #### Elementary Sequences (8) Leading asymptotic term for a polynomial sequence: ```wl In[1]:= DiscreteAsymptotic[n ^ 4 + 5n ^ 3 + 3n - 11, n -> ∞] Out[1]= n^4 ``` Plot the sequence and the approximation: ```wl In[2]:= DiscretePlot[{n ^ 4 + 5n ^ 3 + 3n - 11, n ^ 4}, {n, 1, 20}] Out[2]= [image] ``` --- Rational sequences: ```wl In[1]:= DiscreteAsymptotic[(3n ^ 2 + 5) / (4n ^ 2 + 11), n -> ∞] Out[1]= (3/4) In[2]:= DiscreteAsymptotic[(3n ^ 3 + 5) / (4n ^ 2 + 11), n -> ∞] Out[2]= (3 n/4) In[3]:= DiscreteAsymptotic[(3n + 5) / (4n ^ 2 + 11), n -> ∞] Out[3]= (3/4 n) ``` --- Exponential sequences: ```wl In[1]:= DiscreteAsymptotic[2 ^ n, n -> ∞] Out[1]= 2^n In[2]:= DiscreteAsymptotic[2 ^ n + 3 ^ (-n), n -> ∞] Out[2]= 2^n ``` --- Polynomial exponential sequences: ```wl In[1]:= DiscreteAsymptotic[n ^ 2 2 ^ n, n -> ∞] Out[1]= 2^n n^2 In[2]:= DiscreteAsymptotic[(n ^ 2 + 3n)2 ^ n, n -> ∞] Out[2]= 2^n n^2 ``` --- Rational exponential sequences: ```wl In[1]:= DiscreteAsymptotic[(n ^ 2 / (3n ^ 2 + 1))2 ^ n, n -> ∞] Out[1]= (2^n/3) In[2]:= DiscreteAsymptotic[(n ^ 2 / (3n ^ 3 + 1))2 ^ n, n -> ∞] Out[2]= (2^n/3 n) In[3]:= DiscreteAsymptotic[(5n ^ 4 / (3n ^ 3 + 1))2 ^ n, n -> ∞] Out[3]= (5 2^n n/3) ``` --- Hyperbolic sequences: ```wl In[1]:= DiscreteAsymptotic[Sinh[n], n -> ∞] Out[1]= (E^n/2) In[2]:= DiscreteAsymptotic[ArcSinh[n], n -> ∞] Out[2]= Log[n] ``` --- Logarithmic sequences: ```wl In[1]:= DiscreteAsymptotic[Log[n], n -> ∞] Out[1]= Log[n] In[2]:= DiscreteAsymptotic[(5n ^ 2 / (3n ^ 3 + 1))Log[n], n -> ∞] Out[2]= (5 Log[n]/3 n) In[3]:= DiscreteAsymptotic[(5n ^ 4 / (3n ^ 3 + 1))Log[n], n -> ∞] Out[3]= (5/3) n Log[n] ``` --- Q-sequences: ```wl In[1]:= DiscreteAsymptotic[2 ^ n / (1 + 2 ^ n), n -> ∞] Out[1]= 1 In[2]:= DiscreteAsymptotic[4 ^ n / (1 + 2 ^ n), n -> ∞] Out[2]= 2^n In[3]:= DiscreteAsymptotic[2 ^ n / (1 + 4 ^ n), n -> ∞] Out[3]= 2^-n In[4]:= DiscreteAsymptotic[QPolyGamma[2, n], n -> ∞] Out[4]= (Log[n]/2) ``` #### Special Sequences (6) Find the leading asymptotic term for ``Fibonacci`` as $n$ approaches ``Infinity`` : ```wl In[1]:= DiscreteAsymptotic[Fibonacci[n], n -> ∞] Out[1]= (GoldenRatio^n/Sqrt[5]) ``` Compare the sequence and the approximation: ```wl In[2]:= Table[{Fibonacci[n], N[%]}, {n, 1, 21, 4}]//TableForm Out[2]//TableForm= | | | | :---- | :------- | | 1 | 0.723607 | | 5 | 4.95967 | | 34 | 33.9941 | | 233 | 232.999 | | 1597 | 1597. | | 10946 | 10946. | ``` --- Leading asymptotic term for ``Pochhammer`` : ```wl In[1]:= DiscreteAsymptotic[Pochhammer[a, n], n -> ∞] Out[1]= (E^-n n^-(1/2) + a + n Sqrt[2 π]/Gamma[a]) ``` ``FactorialPower`` : ```wl In[2]:= DiscreteAsymptotic[FactorialPower[a, n], n -> ∞] Out[2]= E^-n n^-(1/2) - a + n Sqrt[(2/π)] Gamma[1 + a] Sin[(1 + a - n) π] ``` ``Binomial`` : ```wl In[3]:= DiscreteAsymptotic[Binomial[n, k], n -> ∞] Out[3]= (n^k/k!) ``` --- Leading asymptotic term for ``HarmonicNumber``: ```wl In[1]:= DiscreteAsymptotic[HarmonicNumber[n], n -> ∞] Out[1]= Log[n] ``` ``PolyGamma``: ```wl In[2]:= DiscreteAsymptotic[PolyGamma[n], n -> ∞] Out[2]= Log[n] ``` ``Zeta`` : ```wl In[3]:= DiscreteAsymptotic[Zeta[n], n -> ∞] Out[3]= 1 ``` --- Leading asymptotic term for ``StirlingS1`` : ```wl In[1]:= DiscreteAsymptotic[StirlingS1[n, 2], n -> ∞] Out[1]= (-1 + n)! Log[n] ``` ``StirlingS2``: ```wl In[2]:= DiscreteAsymptotic[StirlingS2[n, 2], n -> ∞] Out[2]= ((-1 + E^(1 + n/2) + ProductLog[(1/2) E^(1/2) (-1 - n) (-1 - n)])^2 n! ((1 + n/2) + ProductLog[(1/2) E^(1/2) (-1 - n) (-1 - n)])^-n/2 Sqrt[2 π] Sqrt[(1 + n) (1 - ((1 + n/2) + ProductLog[(1/2) E^(1/2) (-1 - n) (-1 - n)]/-1 + E^(1 + n/2) + ProductLog[(1/2) E^(1/2) (-1 - n) (-1 - n)]))]) ``` Compare with the exact value for $n=200$ : ```wl In[3]:= % /. {n -> 200.`20}//N Out[3]= 8.038022041106578*^59 In[4]:= StirlingS2[200, 2]//N Out[4]= 8.034690221294951*^59 ``` --- Leading asymptotic term for ``BellB`` : ```wl In[1]:= DiscreteAsymptotic[BellB[n], n -> ∞] Out[1]= (E^-1 - n + (n/ProductLog[n]) ((n/ProductLog[n]))^(1/2) + n/Sqrt[n]) ``` Compare with the exact value for $n=200$ : ```wl In[2]:= % /. {n -> 1400.`20}//N Out[2]= 2.818063169512889197521886719615913`15.954589770191005*^2865 In[3]:= BellB[1400]//N Out[3]= 2.592514236389446124590286698966767`15.954589770191005*^2865 ``` --- Leading asymptotic term for ``BernoulliB`` : ```wl In[1]:= DiscreteAsymptotic[BernoulliB[2n], n -> ∞] Out[1]= 4 (-1)^-1 + n E^-2 n n^(1/2) + 2 n π^(1/2) - 2 n ``` ``EulerE`` : ```wl In[2]:= DiscreteAsymptotic[EulerE[2n], n -> ∞] Out[2]= (-1)^n 2^3 + 4 n E^-2 n n^(1/2) + 2 n π^-(1/2) - 2 n ``` #### Sums and Summation Transforms (3) Compute an asymptotic approximation for a definite sum: ```wl In[1]:= DiscreteAsymptotic[Inactive[Sum][1 / (k ^ 2 + a), {k, 0, Infinity}], a -> 0] Out[1]= (1/a) ``` Obtain the same result using ``AsymptoticSum`` : ```wl In[2]:= AsymptoticSum[1 / (k ^ 2 + a), {k, 0, Infinity}, a -> 0] Out[2]= (1/a) ``` --- Compute an asymptotic approximation for the Fibonacci sequence using its generating function: ```wl In[1]:= DiscreteAsymptotic[Inactive[SeriesCoefficient][-(z / (-1 + z + z ^ 2)), {z, 0, n}], n -> ∞] Out[1]= (((2/-1 + Sqrt[5]))^n/Sqrt[5]) In[2]:= % /. {n -> 2000.} Out[2]= 4.2246963333916533769256828272394`12.971223483591896*^417 In[3]:= Fibonacci[2000]//N Out[3]= 4.2246963333923048787067256023415`15.954589770191005*^417 ``` --- Compute a leading asymptotic approximation for an inverse Z transform: ```wl In[1]:= DiscreteAsymptotic[Inactive[InverseZTransform][1 / (z * (-6 + 6 * z ^ 2 + z ^ 3)), z, n], n -> ∞] Out[1]= (Root[-1 - 6*#1 + 6*#1^3 & , 2, 0]^3 - n/-6 + 18 Root[-1 - 6*#1 + 6*#1^3 & , 2, 0]^2) ``` #### Difference Equations (2) Compute an asymptotic approximation for a first-order difference equation: ```wl In[1]:= dr = DifferenceRoot[Function[{\[FormalY], \[FormalN]}, {-(n * \[FormalY][\[FormalN]]) + \[FormalY][1 + \[FormalN]] == 0, \[FormalY][1] == 1}]][n]; In[2]:= DiscreteAsymptotic[dr, n -> Infinity] Out[2]= E^-n n^-(1/2) + n Sqrt[2 π] ``` Obtain the same result using ``AsymptoticRSolveValue`` : ```wl In[3]:= AsymptoticRSolveValue[{y[n + 1] - n * y[n] == 0, y[1] == 1}, y[n], n -> ∞] Out[3]= E^-n n^-(1/2) + n Sqrt[2 π] ``` --- Compute an asymptotic approximation for a higher-order difference equation: ```wl In[1]:= DiscreteAsymptotic[Inactive[RSolveValue][{n ^ 4y[n + 2] == 2n ^ 3(n - 1) y[n + 1] - (n ^ 4 - 2n ^ 3 - 1)y[n], y[1] == 1., y[2] == 3}, y[n], n], n -> ∞, SeriesTermGoal -> 7] Out[1]= 2.53605 (1.  + (0.125215/n^6) + (0.106428/n^5) - (0.00449002/n^4) - (0.131289/n^3) - (0.22/n^2) - (0.25/n)) ``` ### Options (1) #### SeriesTermGoal (1) By default, ``DiscreteAsymptotic`` returns the leading term in the asymptotic expansion: ```wl In[1]:= DiscreteAsymptotic[n!, n -> ∞] Out[1]= E^-n n^(1/2) + n Sqrt[2 π] In[2]:= DiscreteAsymptotic[n!, n -> ∞, SeriesTermGoal -> 1] Out[2]= E^-n n^(1/2) + n Sqrt[2 π] ``` Use ``SeriesTermGoal`` to obtain more terms from the expansion: ```wl In[3]:= DiscreteAsymptotic[n!, n -> ∞, SeriesTermGoal -> 3] Out[3]= E^-n (1 + (1/288 n^2) + (1/12 n)) n^(1/2) + n Sqrt[2 π] ``` Obtain the same result using a list specification: ```wl In[4]:= DiscreteAsymptotic[n!, {n, ∞, 3}] Out[4]= E^-n (1 + (1/288 n^2) + (1/12 n)) n^(1/2) + n Sqrt[2 π] ``` ### Applications (3) Find the leading asymptotic term for ``Prime`` as $n$ approaches ``Infinity`` : ```wl In[1]:= DiscreteAsymptotic[Prime[n], n -> ∞] Out[1]= n Log[n] ``` Plot the sequence and the approximation: ```wl In[2]:= DiscretePlot[{Prime[n], %}, {n, 100, 1000, 10}, PlotLegends -> {Prime[n], n Log[n]}] Out[2]= [image] ``` Compare the numerical values of the sequence and its approximation: ```wl In[3]:= {Prime[1000], 1000 Log[1000.]} Out[3]= {7919, 6907.76} ``` The ratio of the sequence and its leading asymptotic approaches ``1`` as $n$ approaches ``Infinity`` : ```wl In[4]:= DiscreteLimit[Prime[n] / (n Log[n]), n -> ∞] Out[4]= 1 ``` --- Compute an asymptotic approximation for a binomial sum: ```wl In[1]:= DiscreteAsymptotic[Sum[Binomial[n, k] ^ 3, {k, 0., n}], n -> ∞] Out[1]= (3.3393986352596523*^133 2.^-445. + 3. n (1.  - (0.333333/n))/n) ``` Compare the approximate and exact values for $n=1000$ : ```wl In[2]:= % /. {n -> 1000.} Out[2]= 4.5202494066862272947783138353279`15.954589770191005*^899 In[3]:= Sum[Binomial[1000, k] ^ 3, {k, 0, 1000}]//N Out[3]= 4.52024229612002556338680147777922`15.954589770191005*^899 ``` --- Compute the leading-order asymptotic term for the Apéry sequence, which satisfies the following linear second-order difference equation: ```wl In[1]:= apeqn = (n + 2) ^ 3 u[n + 2] - (34 n ^ 3 + 153 n ^ 2 + 231 n + 117)u[n + 1] + (n + 1) ^ 3 u[n] == 0; ``` Obtain the leading asymptotic term: ```wl In[2]:= sol[n_] = DiscreteAsymptotic[Inactive[RSolveValue][apeqn, u[n], n], n -> Infinity] Out[2]= ((17 + 12 Sqrt[2])^n C[2]/n^3 / 2) ``` Assign a value to C[2] based on the defining sum for the sequence: ```wl In[3]:= tsol[n_] := Sum[Binomial[n, k] ^ 2 Binomial[n + k, k] ^ 2, {k, 0, n}] In[4]:= Solve[sol[1000] == tsol[1000], C[2]]//N[#, 20]&//Chop Out[4]= {{C[2] -> 0.21995168851940357855}} ``` Obtain an approximate value for a member of the sequence: ```wl In[5]:= sol[10000] /. %[[1]] //N[#, 20]& Out[5]= 2.342830206533991512399013847774535159085281219`20.*^15304 ``` Compare with the corresponding Apéry number: ```wl In[6]:= tsol[10000]//N Out[6]= 2.3437128911923108519528343`15.954589770191005*^15304 ``` ### Properties & Relations (2) The result from ``DiscreteAsymptotic`` is asymptotically equivalent to the sequence: ```wl In[1]:= DiscreteAsymptotic[(n ^ 2 + 3) / (5n ^ 3 + 11), n -> ∞] Out[1]= (1/5 n) ``` Use ``AsymptoticEquivalent`` to verify the result: ```wl In[2]:= AsymptoticEquivalent[(n ^ 2 + 3) / (5n ^ 3 + 11), 1 / (5n), n -> ∞] Out[2]= True ``` --- ``DiscreteAsymptotic`` describes the behavior of a sequence for large values of $n$: ```wl In[1]:= DiscreteAsymptotic[2 ^ n / (1 + 4 ^ n), n -> ∞] Out[1]= 2^-n In[2]:= % /. {n -> 500.} Out[2]= 3.054936363499605`*^-151 ``` ``DiscreteLimit`` describes the behavior of the sequence at ``Infinity`` : ```wl In[3]:= DiscreteLimit[2 ^ n / (1 + 4 ^ n), n -> ∞] Out[3]= 0 ``` ## See Also * [`Asymptotic`](https://reference.wolfram.com/language/ref/Asymptotic.en.md) * [`Series`](https://reference.wolfram.com/language/ref/Series.en.md) * [`DiscreteLimit`](https://reference.wolfram.com/language/ref/DiscreteLimit.en.md) * [`AsymptoticSum`](https://reference.wolfram.com/language/ref/AsymptoticSum.en.md) * [`AsymptoticProduct`](https://reference.wolfram.com/language/ref/AsymptoticProduct.en.md) * [`AsymptoticRSolveValue`](https://reference.wolfram.com/language/ref/AsymptoticRSolveValue.en.md) * [`Sum`](https://reference.wolfram.com/language/ref/Sum.en.md) * [`Product`](https://reference.wolfram.com/language/ref/Product.en.md) * [`RSolveValue`](https://reference.wolfram.com/language/ref/RSolveValue.en.md) * [`SeriesCoefficient`](https://reference.wolfram.com/language/ref/SeriesCoefficient.en.md) * [`InverseZTransform`](https://reference.wolfram.com/language/ref/InverseZTransform.en.md) * [`Prime`](https://reference.wolfram.com/language/ref/Prime.en.md) ## Related Guides * [`Asymptotics`](https://reference.wolfram.com/language/guide/Asymptotics.en.md) ## History * [Introduced in 2020 (12.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn121.en.md)