Theory of One-Dimensional Vlasov-Maxwell Equilibria by Oliver Allanson

Author:Oliver Allanson
Language: eng
Format: epub
ISBN: 9783319975412
Publisher: Springer International Publishing

for and . By rearranging the order of summation, can be written

where is the floor function, denoting the greatest integer less than or equal to x. Recognising an upper bound in the expression for ;

gives

Hence we now have an upper bound on for and we know that , and so is bounded above by . Note also that . Hence, each term in our series for is bounded above by a series known to converge for all according to

So by the comparison test, we can now say that is a convergent series. Hence the representation of the DF in Eq. (3.32) is convergent.

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