Show that
That's right. I just happened to find out this interesting integral when I was doing the line integral posted here today by transgalactic, using both line integral and double integral.
http://www.mathhelpforum.com/math-he...-question.html
Now you have got this: , can you go ahead and find
To me, that's the beauty of this integral. You can continue the differentiation to find and you can also find the curve is a beauty as well in when you plot it out.
I am sure NonCommAlg has a solution for , so can you please share with us?
Additional question is: What happens if is zero or negative. Is there a closed-form solution. I guess not, if so, how about if . I don't have a solution for this. If you have, can you please share.
(*)
Let me give an example showing how the previous equality is used to get the summation of an important series.
[/COLOR][/COLOR] (1)
According to the equality,
(2)
Therefore,
(3)
Taylor's series,
(4)
Plug (3) into (4),
(5)
Multiply on both sides,
(6)
Finally, the series:
(7)
Some Special cases:
(a)
(8)
(b)
(9)
(c)
(10) (which is within everybody's expectation)
By Luobo
08/14/2009