Taylor series of elementary functions about selected points

f(x+h) = f(x) + h f'(x) + h^2/2! f''(x) + h^3/3! f'''(x) + ...


taylor(sqrt(1+x),x,0,11);

                 2    3      4      5       6
            X   X    X    5 X    7 X    21 X
        1 + - - -- + -- - ---- + ---- - -----
            2   8    16   128    256    1024

       7        8        9         10         11
   33 X    429 X    715 X    2431 X     4199 X
 + ----- - ------ + ------ - -------- + --------
   2048    32768    65536    262144     524288

 
taylor(log(1+x),x,0,11);

                 2   3    4    5    6    7    8    9    10   11
                X   X    X    X    X    X    X    X    X    X
            X - - + -- - -- + -- - -- + -- - -- + -- - -- + ---
                2   3    4    5    6    7    8    9    10   11


taylor(exp(x),x,0,11);

                 2    3    4    5     6      7      8        9        10
                X    X    X    X     X      X      X        X        X
        1 + X + -- + -- + -- + --- + --- + ---- + ----- + ------ + -------
                2    6    24   120   720   5040   40320   362880   3628800

                                                                   11
                                                                  X
                                                              + -------- 
                                                                39916800
taylor(sin(x),x,0,11);

                     3    5      7       9        11
                    X    X      X       X        X
                X - -- + --- - ---- + ------ - -------- 
                    6    120   5040   362880   39916800

taylor(cos(x),x,0,11);

                       2    4    6      8        10
                      X    X    X      X        X
                  1 - -- + -- - --- + ----- - ------- 
                      2    24   720   40320   3628800

taylor(tan(x),x,0,11);

                     3      5       7       9         11
                    X    2 X    17 X    62 X    1382 X
                X + -- + ---- + ----- + ----- + -------- 
                    3     15     315    2835     155925

taylor(cot(x),x,0,11);

                       3      5     7       9          11
              1   X   X    2 X     X     2 X     1382 X
              - - - - -- - ---- - ---- - ----- - --------- 
              X   3   45   945    4725   93555   638512875

taylor(asin(x),x,0,11);

                      3      5      7       9       11
                     X    3 X    5 X    35 X    63 X
                 X + -- + ---- + ---- + ----- + ------ 
                     6     40    112    1152     2816

taylor(acos(x),x,0,11);

                         3      5      7       9       11
              %PI       X    3 X    5 X    35 X    63 X
              --- - X - -- - ---- - ---- - ----- - ------ 
               2        6     40    112    1152     2816

taylor(atan(x),x,0,11);

                           3    5    7    9    11
                          X    X    X    X    X
                      X - -- + -- - -- + -- - --- 
                          3    5    7    9    11

taylor(acot(x),x,0,11);

                              3    5    7    9    11
                   %PI       X    X    X    X    X
                   --- - X + -- - -- + -- - -- + --- 
                    2        3    5    7    9    11

taylor(sinh(x),x,0,11);

                     3    5      7       9        11
                    X    X      X       X        X
                X + -- + --- + ---- + ------ + -------- 
                    6    120   5040   362880   39916800

taylor(cosh(x),x,0,11);

                       2    4    6      8        10
                      X    X    X      X        X
                  1 + -- + -- + --- + ----- + ------- 
                      2    24   720   40320   3628800

taylor(tanh(x),x,0,11);

                     3      5       7       9         11
                    X    2 X    17 X    62 X    1382 X
                X - -- + ---- - ----- + ----- - -------- 
                    3     15     315    2835     155925

taylor(coth(x),x,0,11);

                       3      5     7       9          11
              1   X   X    2 X     X     2 X     1382 X
              - + - - -- + ---- - ---- + ----- - --------- 
              X   3   45   945    4725   93555   638512875

taylor(asinh(x),x,0,11);

                      3      5      7       9       11
                     X    3 X    5 X    35 X    63 X
                 X - -- + ---- - ---- + ----- - ------ 
                     6     40    112    1152     2816

taylor(acosh(x),x,0,11);

                               3           5           7            9
           %I %PI          %I X    (3 %I) X    (5 %I) X    (35 %I) X
         - ------ + %I X + ----- + --------- + --------- + ----------
             2               6        40          112         1152

                                                                      11
                                                             (63 %I) X
                                                           + ----------- 
                                                                2816

taylor(atanh(x),x,0,11);

                           3    5    7    9    11
                          X    X    X    X    X
                      X + -- + -- + -- + -- + --- 
                          3    5    7    9    11

taylor(acoth(x),x,0,11);

                                3    5    7    9    11
                  %PI %I       X    X    X    X    X
                  ------ + X + -- + -- + -- + -- + --- 
                    2          3    5    7    9    11