Comment #19 on issue 1028 by trel...@psu.edu: Sum().doit() doesn't do it
http://code.google.com/p/sympy/issues/detail?id=1028
These seem to work now:
In [1]: Sum(binomial(n,k)/binomial(2*n,k)/factorial(k)*(2*x)**k,(k,0,n))
Out[1]:
n
_____
\ `
\ k /n\
\ (2*x) *| |
\ \k/
) ----------
/ /2*n\
/ | |*k!
/ \ k /
/____,
k = 0
In [2]: _.doit()
Out[2]:
-2*n - 1 n + 1/2 n + 1/2 x
2 *2 *x *e *besseli(-n - 1/2, x)*gamma(-n + 1/2)
In [3]: Sum(binomial(n,k)*x**k*y**(n-k),(k,0,n))
Out[3]:
n
___
\ `
\ k -k + n /n\
) x *y *| |
/ \k/
/__,
k = 0
In [4]: _.doit()
Out[4]:
/ n
| n /x \ / / |x| \ /|x| \
| y *|- + 1| for Or|And|-n >= 1, |-| < 1|, And||-| <= 1, -n < 0|,
An
| \y / \ \ |y| / \|y| /
|
| n
< ___
| \ `
| \ k -k n /n\
| ) x *y *y *| | otherwise
| / \k/
| /__,
\k = 0
/ |x| -x \\
d|-n >= 0, |-| <= 1, -n < 1, -- != 1||
\ |y| y //
In [5]: Sum(1/k - 1/(k+1),(k,1,n))
Out[5]:
n
___
\ `
\ / 1 1\
) |- ----- + -|
/ \ k + 1 k/
/__,
k = 1
In [6]: _.doit()
Out[6]:
1
1 - -----
n + 1
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