\hVSSKJr SSKJr SSKJr SSKJr SSKJ r SSK J r SSK J r SS KJr SS KJr SS KJrJr SS KJr SS KJrJr SSKJr SSKJr SSKJr SSK J!r!J"r"J#r#J$r$ SSK%J&r& SSK'J(r(J)r) SSK*J+r+ SSK,J-r-J.r. SSK/J0r0 SSK1J2r2J3r3 SSK4J5r5 SSK6J7r7 SSK8J9r9 SSK:J;r; SSKr> SSK?J@r@ SSKAJBrBJCrC SSKDJErEJFrFJGrG SS KHJIrI SS!KJJKrK SS"KLJMrM SS#KNJOrO SS$KPJQrQ SS%KRJSrSJTrT SS&KUJVrV SS'KWrW"S(S)\ \5rXS*rYS+rZS,r[S-r\S.r]S/r^S0r_S1r`S2raS3rbS4rcg')5) annotations) is_decreasing)AccumulationBounds)ExprWithIntLimits) AddWithLimits) gosper_sum)Expr)Add)Tuple) Derivativeexpand)Mul)Float_illegal)Eq)S)ordered)DummyWildSymbolsymbols) factorial) bernoulliharmonic)re)explog) Piecewise)cotcsc)hyper)KroneckerDelta)zeta)Integral)AndNot)apart)PolynomialErrorPolificationFailed)parallel_poly_from_exprPolyfactor)together) limit_seq)O)residue)Contains) FiniteSetInterval)siftNc\rSrSr%SrSrS\S'SrSrSr S r S r S r S r S rSrSrSrSrSrSSjrSrSrSrg)Sum*a Represents unevaluated summation. Explanation =========== ``Sum`` represents a finite or infinite series, with the first argument being the general form of terms in the series, and the second argument being ``(dummy_variable, start, end)``, with ``dummy_variable`` taking all integer values from ``start`` through ``end``. In accordance with long-standing mathematical convention, the end term is included in the summation. Finite sums =========== For finite sums (and sums with symbolic limits assumed to be finite) we follow the summation convention described by Karr [1], especially definition 3 of section 1.4. The sum: .. math:: \sum_{m \leq i < n} f(i) has *the obvious meaning* for `m < n`, namely: .. math:: \sum_{m \leq i < n} f(i) = f(m) + f(m+1) + \ldots + f(n-2) + f(n-1) with the upper limit value `f(n)` excluded. The sum over an empty set is zero if and only if `m = n`: .. math:: \sum_{m \leq i < n} f(i) = 0 \quad \mathrm{for} \quad m = n Finally, for all other sums over empty sets we assume the following definition: .. math:: \sum_{m \leq i < n} f(i) = - \sum_{n \leq i < m} f(i) \quad \mathrm{for} \quad m > n It is important to note that Karr defines all sums with the upper limit being exclusive. This is in contrast to the usual mathematical notation, but does not affect the summation convention. Indeed we have: .. math:: \sum_{m \leq i < n} f(i) = \sum_{i = m}^{n - 1} f(i) where the difference in notation is intentional to emphasize the meaning, with limits typeset on the top being inclusive. Examples ======== >>> from sympy.abc import i, k, m, n, x >>> from sympy import Sum, factorial, oo, IndexedBase, Function >>> Sum(k, (k, 1, m)) Sum(k, (k, 1, m)) >>> Sum(k, (k, 1, m)).doit() m**2/2 + m/2 >>> Sum(k**2, (k, 1, m)) Sum(k**2, (k, 1, m)) >>> Sum(k**2, (k, 1, m)).doit() m**3/3 + m**2/2 + m/6 >>> Sum(x**k, (k, 0, oo)) Sum(x**k, (k, 0, oo)) >>> Sum(x**k, (k, 0, oo)).doit() Piecewise((1/(1 - x), Abs(x) < 1), (Sum(x**k, (k, 0, oo)), True)) >>> Sum(x**k/factorial(k), (k, 0, oo)).doit() exp(x) Here are examples to do summation with symbolic indices. You can use either Function of IndexedBase classes: >>> f = Function('f') >>> Sum(f(n), (n, 0, 3)).doit() f(0) + f(1) + f(2) + f(3) >>> Sum(f(n), (n, 0, oo)).doit() Sum(f(n), (n, 0, oo)) >>> f = IndexedBase('f') >>> Sum(f[n]**2, (n, 0, 3)).doit() f[0]**2 + f[1]**2 + f[2]**2 + f[3]**2 An example showing that the symbolic result of a summation is still valid for seemingly nonsensical values of the limits. Then the Karr convention allows us to give a perfectly valid interpretation to those sums by interchanging the limits according to the above rules: >>> S = Sum(i, (i, 1, n)).doit() >>> S n**2/2 + n/2 >>> S.subs(n, -4) 6 >>> Sum(i, (i, 1, -4)).doit() 6 >>> Sum(-i, (i, -3, 0)).doit() 6 An explicit example of the Karr summation convention: >>> S1 = Sum(i**2, (i, m, m+n-1)).doit() >>> S1 m**2*n + m*n**2 - m*n + n**3/3 - n**2/2 + n/6 >>> S2 = Sum(i**2, (i, m+n, m-1)).doit() >>> S2 -m**2*n - m*n**2 + m*n - n**3/3 + n**2/2 - n/6 >>> S1 + S2 0 >>> S3 = Sum(i, (i, m, m-1)).doit() >>> S3 0 See Also ======== summation Product, sympy.concrete.products.product References ========== .. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 https://dl.acm.org/doi/10.1145/322248.322255 .. [2] https://en.wikipedia.org/wiki/Summation#Capital-sigma_notation .. [3] https://en.wikipedia.org/wiki/Empty_sum z tuple[tuple[Symbol, Expr, Expr]]limitsc[R"X/UQ70UD6n[US5(dU$[SUR55(a [ S5eU$)Nr:c3V# UHn[U5S:g=(d SU;v M! g7f)N)len).0ls Q/var/www/auris/envauris/lib/python3.13/site-packages/sympy/concrete/summations.py Sum.__new__..s$<As1v{'dai's')z/Sum requires values for lower and upper bounds.)r__new__hasattranyr: ValueError)clsfunctionr assumptionsobjs rArD Sum.__new__sT##CKGK{KsH%%J << < <NO O c^URR(dUR(aggNT)rIis_zerohas_empty_sequenceselfs rA _eval_is_zeroSum._eval_is_zeros! == D$;$;%&>%&G==,, ,'H !rMcrUR(a&URSLaURR$ggr[)r\r]rI is_negativerRs rA_eval_is_negativeSum._eval_is_negativerarMc`UR(aURR(agggrO)r\rI is_finiterRs rA_eval_is_finiteSum._eval_is_finites"  ! !dmm&=&='> !rMc URSS5(aURR"S0UD6nO URn0nURHn[ U5nU(dMXSUS'M U(aUR 5VVs0sHupgXv_M nnnUR U5R"S0UD6n [U [5(a,[U V s/sHoR U5PM sn 5n U $U bU R U5n U $Un U $URR(a0UR5n X :waU R5$[U5$[UR5HupU upnX- nUS:Xa[Rs $UR(aUR (a US-US- pU*n[#X*X45nUcOX R:XaUR%X*X45nUbUs $Us $UR&"U/URU SQ76s $UnM URSS5(a'[U[(5(dUR"S0UD6$U$s snnfs sn f)NdeepTrrr9)getrIdoitr:)_dummy_with_inherited_properties_concreteitemsxreplace isinstancetuple is_Matrixr_eval_matrix_sum enumeraterZero is_integerrceval_sumeval_zeta_functionfuncr)rShintsfrepsxabdkvundodidiexpandednlimitabdifnewf zeta_functions rArnSum.doits3 99VT " " ""+U+A A;;C9#>Aq SV  %)ZZ\2\TQAD\D2--%**3U3C#u%%s;s!ZZ-s;< J ll4(JJ == " "{{}H}}&#D) )!$++.HAGA!%Cbyvv ~~#//1ua!e1BA1y)D| %$($;$;A1y$IM$0,,K99Q9QR99A'/* 99VT " "a++vv&[3 1` and `q > 0` NwexcludeyzrT) is_comparablerzrOneInfinityrmatchrr$r&r'r2 Naturals0r) rSr}r:rrrrrrresultcoeffsqs rArzSum.eval_zeta_functionsa ??q15**1!aee)QY.GH H AJJ  sQC($sQC*@$sUVTWBXaWWaeaiaR01 16 1 VY..E A FI%)Aed1j0!#hr1;;&?"@"Q%!%%-PR"D\+ + 2rMc[U[5(aXR;a[R$UR [ UR5p2URS5nU(aUR"U/UQ76nUupVnXR;dXR;ag[X!SS9nURX5n U $)aA Differentiate wrt x as long as x is not in the free symbols of any of the upper or lower limits. Explanation =========== Sum(a*b*x, (x, 1, a)) can be differentiated wrt x or b but not `a` since the value of the sum is discontinuous in `a`. In a case involving a limit variable, the unevaluated derivative is returned. rlNT)evaluate) rrr free_symbolsrrwrIlistr:popr{r ) rSxr}r:r_rrdfrvs rA_eval_derivativeSum._eval_derivative's" a Q.?.?%?66MMM4 #46 2  !%f%Aa  !~~"5 t , YYr ! rMcURSup4nURXU-5n[UR5S:XaURSnOUR"URSS6n[ XsUS-U45R 5$)Nrlrr)argssubsr>r{r7rn)rSrsteprrupper new_upperr}s rA_eval_difference_deltaSum._eval_difference_deltaJsyiim eJJqd(+ tyy>Q  ! A 499Sb>*A1%!)Y/05577rMc URnURSS5(aUR"S0UD6n[R"[ U55n/n/nUHnUR [5(a[R"[ U55n/nUHLn [U [5(a#URU R"S0UD65 M;URU 5 MN UR[U65 MURU5 M SSK J n Jn [U "U5/UQ76n U "XRS9$)NrkTr) factor_sum sum_combine)r:r9)rIrmsimplifyr make_argsrhasr7rrrappend_eval_simplifysympy.simplify.simplifyrrr:) rSkwargsrItermss_to_ttermsubterms out_termssubtermrrrs rArSum._eval_simplifyUs== ::fd # #((262H fX./Dxx}}==6 'G!'3//!(()?)?)I&)IJ"((1 ( 3 ?+ 4 ', D[%,,&55rMc P^)^*^+^,^-[S[S9upnURSSm,URSSm*URSSm-URR 5n[ UR 5S:a [S5eT*R(a!T-R(a[R$T*[RLaT-[RLa][UT,S[R45R5=(a+ [UT,[RS45R5$SSKJn U"UR!T,T,*055nT-*m*[Rm-[#T,R$SSS 9nUR!T,U05nUm,['T*T-5m)UR((awUR*HWupxUS:Xd-UR-5R.[RLdM8[UT,T*T-45n U R5s $ [R$[1UT,5n U bU R2S La[R4$[1[7U5T,5n U bU R2S La[R4$[9UT,[R45n U R:R=T,U-5n U b0XS :a[R$XS :a[R4$U R:R=ST,U-[?ST,- 5U--[?[?ST,- 5*5U--- 5=(dP U R:R=ST,U-[?ST,- 5*U--[?[?ST,- 5*5U--- 5nUbcXS:d;XS:XaXS:d+XXs=:XaS:Xa*O [R4$XS:a[R$[R4$[1T,U-T,5nUb'UR@(aUS:a[R4$UR!T,T,S-05nSS K!J"n SSK#J$n U"U"UU- 55n[1UT,5nUbbUR@(aQU[RJLa>[7U5S:a[R4$[7U5S:a[R$US:Xa{T,UURMT,T,S-5- S- -nURO5n[1UT,5n U b=U R@(a,U S:a[R$U S:a[R4$[1[7U5ST,- -T,5nUb=UR@(a,US:a[R$US:a[R4$UR=[RPT,U--U-5nUURST,5(d$[UUUT)5(a[R$S nSSK+J,n U"UR[T,5T,T)5nU(dT)nOP[]U[^5(a;UR.R@(a ['UR.T)R.5nUbj[UUU5(d[UU*U5(aG[aUT,T*T-45nURc5nUR@(a[UR5$U R:Rd(aU R:R*n[gU5n[#SSS9m+U)U+U,4SjnU*U,U-4Sjn [iS[ U55Hsn![jRl"UU!5HUn"U[gU"5- n#[oU"6n$[oU#6n%[UU$T)5(aU"U%5n&U&bU&s s $U "U$U%5n'U'cMQU's s $ Mu URSSn(UR!T,U(05n[SU-5e![a GNf=f![a GNf=f![a GNf=f![a S nGNf=f![a GNf=f![a GNf=f![a GNf=f)a Checks for the convergence of a Sum. Explanation =========== We divide the study of convergence of infinite sums and products in two parts. First Part: One part is the question whether all the terms are well defined, i.e., they are finite in a sum and also non-zero in a product. Zero is the analogy of (minus) infinity in products as :math:`e^{-\infty} = 0`. Second Part: The second part is the question of convergence after infinities, and zeros in products, have been omitted assuming that their number is finite. This means that we only consider the tail of the sum or product, starting from some point after which all terms are well defined. For example, in a sum of the form: .. math:: \sum_{1 \leq i < \infty} \frac{1}{n^2 + an + b} where a and b are numbers. The routine will return true, even if there are infinities in the term sequence (at most two). An analogous product would be: .. math:: \prod_{1 \leq i < \infty} e^{\frac{1}{n^2 + an + b}} This is how convergence is interpreted. It is concerned with what happens at the limit. Finding the bad terms is another independent matter. Note: It is responsibility of user to see that the sum or product is well defined. There are various tests employed to check the convergence like divergence test, root test, integral test, alternating series test, comparison tests, Dirichlet tests. It returns true if Sum is convergent and false if divergent and NotImplementedError if it cannot be checked. References ========== .. [1] https://en.wikipedia.org/wiki/Convergence_tests Examples ======== >>> from sympy import factorial, S, Sum, Symbol, oo >>> n = Symbol('n', integer=True) >>> Sum(n/(n - 1), (n, 4, 7)).is_convergent() True >>> Sum(n/(2*n + 1), (n, 1, oo)).is_convergent() False >>> Sum(factorial(n)/5**n, (n, 1, oo)).is_convergent() False >>> Sum(1/n**(S(6)/5), (n, 1, oo)).is_convergent() True See Also ======== Sum.is_absolutely_convergent sympy.concrete.products.Product.is_convergent zp q r)rHrrrzNconvergence checking for more than one symbol containing series is not handledrTintegerpositiveNFrlcombsimp)powsimp)solvesetm)rc>[[UTTRT45R5T5nUb"UR(a[ R $gg![a gf=fN)r/r7infrnrgrtrueNotImplementedError)g_ning_valintervalrsyms rA_dirichlet_test*Sum.is_convergent.._dirichlet_testxsb'C#x||Q1G(H(M(M(OQRSG*w/@/@ vv 0A**sAA A'&A'cL>[UT5nUbUR(d=[U[5(a\URUR - R(a3[ UTTT45R5(a[R$gggg![a gf=fr) r/rgrrrmaxminr7is_absolutely_convergentrrr)g1_ng2_nlim_val lower_limitr upper_limits rA_bounded_convergent_test3Sum.is_convergent.._bounded_convergent_tests'c2G*0A0A"7,>??$[[7;;6AA"4#{K)HIbbdd'(vv  eB@+ +sBB B#"B#zFThe algorithm to find the Sum convergence of %s is not yet implemented)8rrr:rIrr>rrrgrrNegativeInfinityrr7 is_convergentrrqrnamer4 is_Piecewiseras_setsupr/rPfalseabsr0exprrr is_numbersympy.simplify.combsimprsympy.simplify.powsimprNaNr gammasimp NegativeOnerrsympy.solvers.solvesetrdiffrrr3r%rnis_Mulsetrange itertools combinationsr).rSprr sequence_termrsym_r{condrr lim_val_absorder p_series_test n_log_testlim_compnext_sequence_termrrratio lim_ratiotest_val lim_evaluateddict_valcheck_intervalrmaxima integral_valintegral_val_evaluatedrargsetrrra_tupleb_seta_nb_ndirichbc_test_symrrrrrs. @@@@@rArSum.is_convergent|sT't,akk!nQkk!nQ' kk!nQ'  ..0 })) *Q .%'IJ J  [%:%:66M !,, ,ajj(=31::*>?MMOYMC1C1CQ+GHVVXY 8$]%;%;S3$K%HIM&,K**KSXXtd;%..T{; K5  % %+00 4<4;;=#4#4 #BD3 ["ABA??,, 1 66M  s3G"w%'?ww #C $6+>%+Gww-#qzz!23 ((a0  $"$vv 2%wwjj&&q#q&QsUQ*>sC#J;?OQR?R*R'STXjj&&q#q&3qu:+1A*A#s1S5zkBRTUBU*U'VW   ! !!# (9*-41477N:D9J66M77N  ]!2C8H#(:(:x!|ww +33S#'NC42!3M!ABC !%-I$)<)<RSRWRWAWy>A%77Ny>A%66M >M&++Cq9:<=>?H))+H #Hc2&7+<+<{ vv { ww  %c-&81S5&A3GM(]-D-D 1$66M 1$77N !&&q}}sQw'?'AB{s## hqk8(L(L66M3-,,S13A%N  * *vzz/C/C%fjj(,,?N  & =. 9 9 =.. 9 9'!Ck#B D -9->->-@*-77 !7!A!ABB82 ::  ::??DYFc4(A  1c&k*(55dA>G"S\1Ew-Cu+C$S(33!0!5!-#)M6sC@G*& ?+{{1~a %..T{; !#;>K#MN NU#   #   <#   # I '  #   2+s2-`/!6a8a?Aa"a"%5a5a52Ab7b#5b/ `=<`= a a aa" a21a25 bb bb b%$b%cp[[UR5UR5R 5$)a Checks for the absolute convergence of an infinite series. Same as checking convergence of absolute value of sequence_term of an infinite series. References ========== .. [1] https://en.wikipedia.org/wiki/Absolute_convergence Examples ======== >>> from sympy import Sum, Symbol, oo >>> n = Symbol('n', integer=True) >>> Sum((-1)**n, (n, 1, oo)).is_absolutely_convergent() False >>> Sum((-1)**n/n**2, (n, 1, oo)).is_absolutely_convergent() True See Also ======== Sum.is_convergent )r7rrIr:rrRs rArSum.is_absolutely_convergents'63t}}%t{{3AACCrMc ^^^[U5n[U5nURn[UR5S:wa [ S5eURSummmTT:S:Xa7TT- S:Xa [ R [ R 4$TS-TS- smmU*n[ R nU(GakTR(a#TR(a[UTT- S-5nU(aURT5(a5[[U5Vs/sHouRTTU-5PM sn6nOURTT5nU(aH[URS55U:n U S:Xa U[U54$U S:XdU[ R 4$Un[SU5HOnURTTU-5n[URS55U:aUS:waU[U54s $Xh- nMQ TT- S-U:XaU[ R 4$TU- m[S5n [!URTU 5U TT45n U(aU R#5n Xk- nUUU4Sjn U "U5upX-S - nUR%T5n[SUS -5HnU "U5unn['S U-5[)S U-5- UU- -nXr:a O~U(a^U(aWURS5nU[ R*La"[ R*[ R*4s $[U5U:a OXh- nUR%TS SS 9nM Xo-[W54$s snf) a Return an Euler-Maclaurin approximation of self, where m is the number of leading terms to sum directly and n is the number of terms in the tail. With m = n = 0, this is simply the corresponding integral plus a first-order endpoint correction. Returns (s, e) where s is the Euler-Maclaurin approximation and e is the estimated error (taken to be the magnitude of the first omitted term in the tail): >>> from sympy.abc import k, a, b >>> from sympy import Sum >>> Sum(1/k, (k, 2, 5)).doit().evalf() 1.28333333333333 >>> s, e = Sum(1/k, (k, 2, 5)).euler_maclaurin() >>> s -log(2) + 7/20 + log(5) >>> from sympy import sstr >>> print(sstr((s.evalf(), e.evalf()), full_prec=True)) (1.26629073187415, 0.0175000000000000) The endpoints may be symbolic: >>> s, e = Sum(1/k, (k, a, b)).euler_maclaurin() >>> s -log(a) + log(b) + 1/(2*b) + 1/(2*a) >>> e Abs(1/(12*b**2) - 1/(12*a**2)) If the function is a polynomial of degree at most 2n+1, the Euler-Maclaurin formula becomes exact (and e = 0 is returned): >>> Sum(k, (k, 2, b)).euler_maclaurin() (b**2/2 + b/2 - 1, 0) >>> Sum(k, (k, 2, b)).doit() b**2/2 + b/2 - 1 With a nonzero eps specified, the summation is ended as soon as the remainder term is less than the epsilon. rzMore than 1 limitrTr=Frc>T[RLaURTT5S4$URTT5URTT54$)Nr)rrr)rrrrs rAfpoint#Sum.euler_maclaurin..fpointsBAJJyyA))99Q?DIIaO3 3rMrr)intrIr>r:rGrrw is_Integerr is_polynomialr rrrevalfrr%rnrrrr)rSrreps eval_integralr}rrrtestrIrfafbitermggagb term_evalfrrrs @@@rAeuler_maclaurinSum.euler_maclaurinsV F F MM t{{ q 01 1++a.1a Ed?1uzvvqvv~%q5!a%DAqA FF || 1q519%!//!,,E!H=Hq&&AE*H=>vva|tzz!}-3Dt| #d)|+"em#QVV|+q!A66!QU+D4::a=)C/DAI #d)|+IA % 1uqyA~!&&y FA #J QVVAq\Aq!9 - A  4!  FF1Iq!a%AAYFBQqS>)AaC.0"r':Dut!ZZ] &55!%%<'z?S( IAq!e,A!y#d)##[>sM#cf[U5n[U5H/up4[U[5(aMUR U5X#'M1 Sn/n[UR 5H4up7UnX2;aU*nUSUSS-USS- 4nUR U5 M6 [XPR-/UQ76$)a Reverse the order of a limit in a Sum. Explanation =========== ``reverse_order(self, *indices)`` reverses some limits in the expression ``self`` which can be either a ``Sum`` or a ``Product``. The selectors in the argument ``indices`` specify some indices whose limits get reversed. These selectors are either variable names or numerical indices counted starting from the inner-most limit tuple. Examples ======== >>> from sympy import Sum >>> from sympy.abc import x, y, a, b, c, d >>> Sum(x, (x, 0, 3)).reverse_order(x) Sum(-x, (x, 4, -1)) >>> Sum(x*y, (x, 1, 5), (y, 0, 6)).reverse_order(x, y) Sum(x*y, (x, 6, 0), (y, 7, -1)) >>> Sum(x, (x, a, b)).reverse_order(x) Sum(-x, (x, b + 1, a - 1)) >>> Sum(x, (x, a, b)).reverse_order(0) Sum(-x, (x, b + 1, a - 1)) While one should prefer variable names when specifying which limits to reverse, the index counting notation comes in handy in case there are several symbols with the same name. >>> S = Sum(x**2, (x, a, b), (x, c, d)) >>> S Sum(x**2, (x, a, b), (x, c, d)) >>> S0 = S.reverse_order(0) >>> S0 Sum(-x**2, (x, b + 1, a - 1), (x, c, d)) >>> S1 = S0.reverse_order(1) >>> S1 Sum(x**2, (x, b + 1, a - 1), (x, d + 1, c - 1)) Of course we can mix both notations: >>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(x, 1) Sum(x*y, (x, b + 1, a - 1), (y, 6, 1)) >>> Sum(x*y, (x, a, b), (y, 2, 5)).reverse_order(y, x) Sum(x*y, (x, b + 1, a - 1), (y, 6, 1)) See Also ======== sympy.concrete.expr_with_intlimits.ExprWithIntLimits.index, reorder_limit, sympy.concrete.expr_with_intlimits.ExprWithIntLimits.reorder References ========== .. [1] Michael Karr, "Summation in Finite Terms", Journal of the ACM, Volume 28 Issue 2, April 1981, Pages 305-350 https://dl.acm.org/doi/10.1145/322248.322255 rrr) rrvrrrindexr:rr7rI) rSindices l_indicesrindxer:rr@s rA reverse_orderSum.reverse_order(s|M  +GAdC((#zz$/ , !$++.HAA~B1XuQx!|U1X\: MM!  /1}}$.v..rMcSSKJn URR(a/[ U"[ UR5/UR Q765$g)Nr)Product)sympy.concrete.productsr8rIrWrrr:)rSrrr8s rA_eval_rewrite_as_ProductSum._eval_rewrite_as_Productws93 == ) )ws4==1@DKK@A A *rMN)rrrT)__name__ __module__ __qualname____firstlineno____doc__ __slots____annotations__rDrTrXr_rdrhrnrzrrrrrr-r5r:__static_attributes__r9rMrAr7r7*srBHI ,,. --;z+*!F 8%6NbNH D:h$VM/^BrMr7c:[U/UQ70UD6RSS9$)aF Compute the summation of f with respect to symbols. Explanation =========== The notation for symbols is similar to the notation used in Integral. summation(f, (i, a, b)) computes the sum of f with respect to i from a to b, i.e., :: b ____ \ ` summation(f, (i, a, b)) = ) f /___, i = a If it cannot compute the sum, it returns an unevaluated Sum object. Repeated sums can be computed by introducing additional symbols tuples:: Examples ======== >>> from sympy import summation, oo, symbols, log >>> i, n, m = symbols('i n m', integer=True) >>> summation(2*i - 1, (i, 1, n)) n**2 >>> summation(1/2**i, (i, 0, oo)) 2 >>> summation(1/log(n)**n, (n, 2, oo)) Sum(log(n)**(-n), (n, 2, oo)) >>> summation(i, (i, 0, n), (n, 0, m)) m**3/6 + m**2/2 + m/3 >>> from sympy.abc import x >>> from sympy import factorial >>> summation(x**n/factorial(n), (n, 0, oo)) exp(x) See Also ======== Sum Product, sympy.concrete.products.product F)rk)r7rn)r}rrs rA summationrE}s(d q %7 %f % * * * 66rMc UupEn[[U5Vs/sH+opRXEU-5URXFU- 5-PM- sn6$s snf)a Returns the direct summation of the terms of a telescopic sum Explanation =========== L is the term with lower index R is the term with higher index n difference between the indexes of L and R Examples ======== >>> from sympy.concrete.summations import telescopic_direct >>> from sympy.abc import k, a, b >>> telescopic_direct(1/k, -1/(k+2), 2, (k, a, b)) -1/(b + 2) - 1/(b + 1) + 1/(a + 1) + 1/a )r rr)LRrr:rrrrs rAtelescopic_directrIsK(IQ1 eAhGhq5!AFF1!e$44hG HHGs2AcUup4nUR(dUR(ag[S5nU*RURX3U-55nSnU(a7Xg;a2XvnUR(aURX3U-5U-S:XdSnUc[ S5n SSKJn U "URX3U -5U-U 5=(d /nUV s/sHMoR(dMURX3U -5U-R5R(dMKU PMO nn [U5S:wagUSnUS:a[X[U5X4U45$US:a[XXXE45$g![a gf=fs sn f)z] Tries to perform the summation using the telescopic property. 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[#] http://www.supermath.info/InfiniteSeriesandtheResidueTheorem.pdf .. [#] Asmar N.H., Grafakos L. (2018) Residue Theory. In: Complex Analysis with Applications. Undergraduate Texts in Mathematics. 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