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Finding the 1st, 2nd, 3rd, 4th, 5th, and 6th cumulants in terms of moments

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Find the 1st, 2nd, 3rd, 4th, 5th, and 6th cumulants in terms of moments by using the generating function, showing all intermediate steps.

For the structure I'm trying to follow, please see attached document, also available from

http://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2007/lecture-notes/

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