10 - PRODUTORIO-SOMATORIO-SERIES-MATEMATICAS

502 days ago by jmarcellopereira

SOMATÓRIO, PRODUTÓRIO E SÉRIES MATEMÁTICAS

SOMATÓRIO

var('k') sum(1/k^5, k, 1, 10).n() 
       
1.03690734134469
1.03690734134469
# de 1 até o infinito sum(1/k^5, k, 1, oo).n() 
       
1.03692775514337
1.03692775514337

PRODUTÓRIO

prod([1,2,3]) 
       
6
6
prod([2,4], 5) 
       
40
40
prod((2,4), 5) 
       
40
40
F = factor(1084200) F 
       
2^3 * 3 * 5^2 * 13 * 139
2^3 * 3 * 5^2 * 13 * 139
prod(F) 
       
1084200
1084200
prod? 
       

File: /home/jmarcellopereira/SageMath/src/sage/misc/misc_c.pyx

Type: <type ‘builtin_function_or_method’>

Definition: prod(x, z=None, recursion_cutoff=5)

Docstring:

Return the product of the elements in the list x.

If optional argument z is not given, start the product with the first element of the list, otherwise use z. The empty product is the int 1 if z is not specified, and is z if given.

This assumes that your multiplication is associative; we don’t promise which end of the list we start at.

EXAMPLES:

sage: prod([1,2,34])
68
sage: prod([2,3], 5)
30
sage: prod((1,2,3), 5)
30
sage: F = factor(-2006); F
-1 * 2 * 17 * 59
sage: prod(F)
-2006

AUTHORS:

  • Joel B. Mohler (2007-10-03): Reimplemented in Cython and optimized
  • Robert Bradshaw (2007-10-26): Balanced product tree, other optimizations, (lazy) generator support
  • Robert Bradshaw (2008-03-26): Balanced product tree for generators and iterators

File: /home/jmarcellopereira/SageMath/src/sage/misc/misc_c.pyx

Type: <type ‘builtin_function_or_method’>

Definition: prod(x, z=None, recursion_cutoff=5)

Docstring:

Return the product of the elements in the list x.

If optional argument z is not given, start the product with the first element of the list, otherwise use z. The empty product is the int 1 if z is not specified, and is z if given.

This assumes that your multiplication is associative; we don’t promise which end of the list we start at.

EXAMPLES:

sage: prod([1,2,34])
68
sage: prod([2,3], 5)
30
sage: prod((1,2,3), 5)
30
sage: F = factor(-2006); F
-1 * 2 * 17 * 59
sage: prod(F)
-2006

AUTHORS:

  • Joel B. Mohler (2007-10-03): Reimplemented in Cython and optimized
  • Robert Bradshaw (2007-10-26): Balanced product tree, other optimizations, (lazy) generator support
  • Robert Bradshaw (2008-03-26): Balanced product tree for generators and iterators
taylor? 
       

File: /home/jmarcellopereira/SageMath/local/lib/python2.7/site-packages/sage/calculus/functional.py

Type: <type ‘function’>

Definition: taylor(f, *args)

Docstring:

Expands self in a truncated Taylor or Laurent series in the variable v around the point a, containing terms through (x - a)^n. Functions in more variables are also supported.

INPUT:

  • *args - the following notation is supported
  • x, a, n - variable, point, degree
  • (x, a), (y, b), ..., n - variables with points, degree of polynomial

EXAMPLES:

sage: var('x,k,n')
(x, k, n)
sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6)
-1/720*(45*k^6 - 60*k^4 + 16*k^2)*x^6 - 1/24*(3*k^4 - 4*k^2)*x^4 - 1/2*k^2*x^2 + 1
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1

Taylor polynomial in two variables:

sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
(x - 1)*(y + 1)^3 - 3*(x - 1)*(y + 1)^2 + (y + 1)^3 + 3*(x - 1)*(y + 1) - 3*(y + 1)^2 - x + 3*y + 3

File: /home/jmarcellopereira/SageMath/local/lib/python2.7/site-packages/sage/calculus/functional.py

Type: <type ‘function’>

Definition: taylor(f, *args)

Docstring:

Expands self in a truncated Taylor or Laurent series in the variable v around the point a, containing terms through (x - a)^n. Functions in more variables are also supported.

INPUT:

  • *args - the following notation is supported
  • x, a, n - variable, point, degree
  • (x, a), (y, b), ..., n - variables with points, degree of polynomial

EXAMPLES:

sage: var('x,k,n')
(x, k, n)
sage: taylor (sqrt (1 - k^2*sin(x)^2), x, 0, 6)
-1/720*(45*k^6 - 60*k^4 + 16*k^2)*x^6 - 1/24*(3*k^4 - 4*k^2)*x^4 - 1/2*k^2*x^2 + 1
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1
sage: taylor ((x + 1)^n, x, 0, 4)
1/24*(n^4 - 6*n^3 + 11*n^2 - 6*n)*x^4 + 1/6*(n^3 - 3*n^2 + 2*n)*x^3 + 1/2*(n^2 - n)*x^2 + n*x + 1

Taylor polynomial in two variables:

sage: x,y=var('x y'); taylor(x*y^3,(x,1),(y,-1),4)
(x - 1)*(y + 1)^3 - 3*(x - 1)*(y + 1)^2 + (y + 1)^3 + 3*(x - 1)*(y + 1) - 3*(y + 1)^2 - x + 3*y + 3
var('x,k,n') 
       
taylor ((x-5)^n, x, 0, 2) 
       

%%% FIM SOMATORIO, PRODUTORIO E SERIES MATEMATICAS %%%

sum? 
       

File: /home/jmarcellopereira/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py

Type: <type ‘function’>

Definition: sum(expression, *args, **kwds)

Docstring:

Returns the symbolic sum \sum_{v = a}^b expression with respect to the variable v with endpoints a and b.

INPUT:

  • expression - a symbolic expression
  • v - a variable or variable name
  • a - lower endpoint of the sum
  • b - upper endpoint of the sum
  • algorithm - (default: 'maxima') one of
    • 'maxima' - use Maxima (the default)
    • 'maple' - (optional) use Maple
    • 'mathematica' - (optional) use Mathematica
    • 'giac' - (optional) use Giac

EXAMPLES:

sage: k, n = var('k,n')
sage: sum(k, k, 1, n).factor()
1/2*(n + 1)*n
sage: sum(1/k^4, k, 1, oo)
1/90*pi^4
sage: sum(1/k^5, k, 1, oo)
zeta(5)

Warning

This function only works with symbolic expressions. To sum any other objects like list elements or function return values, please use python summation, see http://docs.python.org/library/functions.html#sum

In particular, this does not work:

sage: n = var('n')
sage: list=[1,2,3,4,5]
sage: sum(list[n],n,0,3)
Traceback (click to the left of this block for traceback)
...
                                
                            

File: /home/jmarcellopereira/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py

Type: <type ‘function’>

Definition: sum(expression, *args, **kwds)

Docstring:

Returns the symbolic sum \sum_{v = a}^b expression with respect to the variable v with endpoints a and b.

INPUT:

  • expression - a symbolic expression
  • v - a variable or variable name
  • a - lower endpoint of the sum
  • b - upper endpoint of the sum
  • algorithm - (default: 'maxima') one of
    • 'maxima' - use Maxima (the default)
    • 'maple' - (optional) use Maple
    • 'mathematica' - (optional) use Mathematica
    • 'giac' - (optional) use Giac

EXAMPLES:

sage: k, n = var('k,n')
sage: sum(k, k, 1, n).factor()
1/2*(n + 1)*n
sage: sum(1/k^4, k, 1, oo)
1/90*pi^4
sage: sum(1/k^5, k, 1, oo)
zeta(5)

Warning

This function only works with symbolic expressions. To sum any other objects like list elements or function return values, please use python summation, see http://docs.python.org/library/functions.html#sum

In particular, this does not work:

sage: n = var('n')
sage: list=[1,2,3,4,5]
sage: sum(list[n],n,0,3)
Traceback (most recent call last):
...
TypeError: unable to convert n to an integer

Use python sum() instead:

sage: sum(list[n] for n in range(4))
10

Also, only a limited number of functions are recognized in symbolic sums:

sage: sum(valuation(n,2),n,1,5)
Traceback (most recent call last):
...
TypeError: unable to convert n to an integer

Again, use python sum():

sage: sum(valuation(n+1,2) for n in range(5))
3

(now back to the Sage sum examples)

A well known binomial identity:

sage: sum(binomial(n,k), k, 0, n)
2^n

The binomial theorem:

sage: x, y = var('x, y')
sage: sum(binomial(n,k) * x^k * y^(n-k), k, 0, n)
(x + y)^n
sage: sum(k * binomial(n, k), k, 1, n)
2^(n - 1)*n
sage: sum((-1)^k*binomial(n,k), k, 0, n)
0
sage: sum(2^(-k)/(k*(k+1)), k, 1, oo)
-log(2) + 1

Another binomial identity (trac ticket #7952):

sage: t,k,i = var('t,k,i')
sage: sum(binomial(i+t,t),i,0,k)
binomial(k + t + 1, t + 1)

Summing a hypergeometric term:

sage: sum(binomial(n, k) * factorial(k) / factorial(n+1+k), k, 0, n)
1/2*sqrt(pi)/factorial(n + 1/2)

We check a well known identity:

sage: bool(sum(k^3, k, 1, n) == sum(k, k, 1, n)^2)
True

A geometric sum:

sage: a, q = var('a, q')
sage: sum(a*q^k, k, 0, n)
(a*q^(n + 1) - a)/(q - 1)

The geometric series:

sage: assume(abs(q) < 1)
sage: sum(a*q^k, k, 0, oo)
-a/(q - 1)

A divergent geometric series. Don’t forget to forget your assumptions:

sage: forget()
sage: assume(q > 1)
sage: sum(a*q^k, k, 0, oo)
Traceback (most recent call last):
...
ValueError: Sum is divergent.

This summation only Mathematica can perform:

sage: sum(1/(1+k^2), k, -oo, oo, algorithm = 'mathematica')     # optional - mathematica
pi*coth(pi)

Use Maple as a backend for summation:

sage: sum(binomial(n,k)*x^k, k, 0, n, algorithm = 'maple')      # optional - maple
(x + 1)^n

Python ints should work as limits of summation (trac ticket #9393):

sage: sum(x, x, 1r, 5r)
15

Note

  1. Sage can currently only understand a subset of the output of Maxima, Maple and Mathematica, so even if the chosen backend can perform the summation the result might not be convertable into a Sage expression.