Series

You can sum a finite series:
> sum(k^2, k=1..5);

                                  55

> sum(k^2, k=1..n);

                         3              2
              1/3 (n + 1)  - 1/2 (n + 1)  + 1/6 n + 1/6

You can find some infinte sum as well:
> sum(1/2^k, k=1.. infinity);

                                  1

Taylor Series

> taylor(exp(x), x=0, 5);

                           2        3         4      5
              1 + x + 1/2 x  + 1/6 x  + 1/24 x  + O(x )

The Taylor series is expanded about x=0, and the first five terms are displayed.
> s := taylor(exp(-x)*cos(x), x=2, 7);

s := exp(-2) cos(2) + (-exp(-2) sin(2) - exp(-2) cos(2)) (x - 2) +

                          2
    exp(-2) sin(2) (x - 2)  +

                                                     3
    (1/3 exp(-2) cos(2) - 1/3 exp(-2) sin(2)) (x - 2)  -

                              4
    1/6 exp(-2) cos(2) (x - 2)  +

                                                       5
    (1/30 exp(-2) sin(2) + 1/30 exp(-2) cos(2)) (x - 2)  -

                               6            7
    1/90 exp(-2) sin(2) (x - 2)  + O((x - 2) )

> sp := convert(s, polynom);

sp := exp(-2) cos(2) + (-exp(-2) sin(2) - exp(-2) cos(2)) (x - 2)

                             2
     + exp(-2) sin(2) (x - 2)

                                                        3
     + (1/3 exp(-2) cos(2) - 1/3 exp(-2) sin(2)) (x - 2)

                                 4
     - 1/6 exp(-2) cos(2) (x - 2)

                                                          5
     + (1/30 exp(-2) sin(2) + 1/30 exp(-2) cos(2)) (x - 2)

                                  6
     - 1/90 exp(-2) sin(2) (x - 2)

> plot({exp(-x)*cos(x), sp}, x=-2..5);

Here, convert command converts the Tayor series approximation into a polynomial .

Partial Derivatives

You can find the partial derivatives:
> diff(x^2*y^3 + exp(x) + ln(y), x);

                                3
                           2 x y  + exp(x)

> diff(x^2*y^3 + exp(x) + ln(y), y);

                               2  2
                            3 x  y  + 1/y

You can find the second partial derivatives:
> diff(diff(x^2*y^3 + cos(x)*sin(y),x),y);

                             2
                        6 x y  - sin(x) cos(y)

> with(linalg);
Warning, new definition for norm
Warning, new definition for trace

[BlockDiagonal, GramSchmidt, JordanBlock, LUdecomp, QRdecomp,

    Wronskian, addcol, addrow, adj, adjoint, angle, augment, backsub,

    band, basis, bezout, blockmatrix, charmat, charpoly, cholesky,

    col, coldim, colspace, colspan, companion, concat, cond, copyinto,

    crossprod, curl, definite, delcols, delrows, det, diag, diverge,

    dotprod, eigenvals, eigenvalues, eigenvectors, eigenvects,

    entermatrix, equal, exponential, extend, ffgausselim, fibonacci,

    forwardsub, frobenius, gausselim, gaussjord, geneqns, genmatrix,

    grad, hadamard, hermite, hessian, hilbert, htranspose, ihermite,

    indexfunc, innerprod, intbasis, inverse, ismith, issimilar,

    iszero, jacobian, jordan, kernel, laplacian, leastsqrs, linsolve,

    matadd, matrix, minor, minpoly, mulcol, mulrow, multiply, norm,

    normalize, nullspace, orthog, permanent, pivot, potential,

    randmatrix, randvector, rank, ratform, row, rowdim, rowspace,

    rowspan, rref, scalarmul, singularvals, smith, stack, submatrix,

    subvector, sumbasis, swapcol, swaprow, sylvester, toeplitz, trace,

    transpose, vandermonde, vecpotent, vectdim, vector, wronskian]

> F :=[x*y*sin(z), x^2*cos(y), z*sqrt(x*y)];

                                  2                1/2
               F := [x y sin(z), x  cos(y), z (x y)   ]

> v :=[x,y,z];

                            v := [x, y, z]

> diverge(F, v);

                               2               1/2
                   y sin(z) - x  sin(y) + (x y)

> curl(F, v);

   [      z x                        z y                          ]
   [1/2 --------, x y cos(z) - 1/2 --------, 2 x cos(y) - x sin(z)]
   [         1/2                        1/2                       ]
   [    (x y)                      (x y)                          ]

> 
> int(int(x*y^2, y=0.. sqrt(1-x)), x=0..1);

                                4/105

> limit((x^2 -y^2)/(x^2 +y^2), {x=0,y=0});

                              undefined

> limit((x + 1/y),{x=0, y=infinity});

                                  0

> 9+