Note the similar formula for (a+b)^n and
the nth derivative with respect to x of
[(x+1)^2]*[e^(2x)].
The nth derivative with respect to x of
e^(2x), at x=0, is 2^n.
At x=0, the
oth derivative of (x+1)^2 is 1,
1st derivative of (x+1)^2 is 2,
2nd derivative of (x+1)^2 is 2,
nth derivative of (x+1)^2 is 0
for n > 2.
Therefore, the 50th derivative with respect
to x of (x+1)^2*e^(2x), at x=0, is
50!
 (1)*(2^50)
0!(500)!
50!
+  (2)*(2^49)
1!(501)!
50!
+  (2)*(2^48)
2!(502)!
= (1)*(1)*(2^50)+(50)*(2)*(2^49)+(25*49)*(2)*(2^48)
= (2^49)*(2 + 100 + 1225)
= 1327*2^49.
Thus, k = 1327 and n = 49.

Posted by Bractals
on 20091218 13:51:20 