(C1) DEFTAYLOR(F(X),X**2+SUM(X**I/(2**I*I!**2), I,4,INF)); (D1) [F] (C2) TAYLOR(%E**SQRT(F(X)),X,0,4); 2 3 4 X 3073 X 12817 X (D2)/R/ 1 + X + -- + ------- + -------- + . . . 2 18432 307200
(C15) ff:taylor(1+x+x^2+x^3,x,0,3); 2 3 (D15)/T/ 1 + X + X + X + . . . (C16) pade(ff,1,1); 1 (D16) [- -----] X - 1 (c1) ff:taylor(-(83787*X^10-45552*X^9-187296*X^8 +387072*X^7+86016*X^6-1507328*X^5 +1966080*X^4+4194304*X^3-25165824*X^2 +67108864*X-134217728) /134217728,x,0,10); (C25) PADE(ff,4,4); (D25) []
There is no rational function of degree 4 numerator/denominator, with this power series expansion. You must in general have degree of the numerator and degree of the denominator adding up to at least the degree of the power series, in order to have enough unknown coefficients to solve.
(C26) PADE(ff,5,5); (D26) [-(520256329*X^5-96719020632*X^4-489651410240*X^3 -1619100813312*X^2 -2176885157888*X-2386516803584) /(47041365435*X^5+381702613848*X^4+1360678489152*X^3 +2856700692480*X^2 +3370143559680*X+2386516803584)]
(C1) VERBOSE:TRUE$ (C2) POWERSERIES(LOG(SIN(X)/X),X,0); Can't expand LOG(SIN(X)) So we'll try again after applying the rule: d / -- (SIN(X)) [ dX LOG(SIN(X)) = I ----------- dX ] SIN(X) / In the first simplification we have returned: / [ I COT(X) dX - LOG(X) ] / INF ==== I1 2 I1 2 I1 \ (- 1) 2 BERN(2 I1) X > ------------------------------ / I1 (2 I1)! ==== I1 = 1 (D2) ------------------------------------- 2
REVERT2(expression,variable,hipower)
also. REVERT only works on expansions around 0.
(C3) TAYLOR((1-Y^2)/(1-X),X,0,3,[Y,A,INF]); 2 2 (D3)/R/ 1 - A - 2 A (Y - A) - (Y - A) 2 2 + (1 - A - 2 A (Y - A) - (Y - A) ) X 2 2 2 + (1 - A - 2 A (Y - A) - (Y - A) ) X 2 2 3 + (1 - A - 2 A (Y - A) - (Y - A) ) X + . . . (C4) TAYLORINFO(D3); (D4) [[Y, A, INF], [X, 0, 3]]
(C1) G(P):=P*4^N/BINOMIAL(2*N,N); N P 4 (D1) G(P) := ---------------- BINOMIAL(2 N, N) (C2) G(N^4); 4 N N 4 (D2) ---------------- BINOMIAL(2 N, N) (C3) NUSUM(D2,N,0,N); 4 3 2 N 2 (N + 1) (63 N + 112 N + 18 N - 22 N + 3) 4 2 (D3) ------------------------------------------------ - ------ 693 BINOMIAL(2 N, N) 3 11 7 (C4) UNSUM(%,N); 4 N N 4 (D4) ---------------- BINOMIAL(2 N, N)