A Simple Expression?
Date: 1/26/96 at 8:32:18 From: Anonymous Subject: Math Problem Hi - I heard that you are the right point of contact for any small or big math problem. My problem is the following: I know that infinity sum a^k/k! = exp(a) k=0 But is it possible to have a simple expression for the same series starting at k=b, that is: infinity sum a^k/k! = ? k=b Thank you for information, Jean-Francois Hellings Project Engineer, SAIT Systems Brussels, Belgium
Date: 8/3/96 at 9:33:47 From: Doctor Jerry Subject: Re: Math Problem Are you asking if there is a simple expression for the tail of the exponential series - if there is a closed form for the series a^b/b!+a^(b+1)/(b+1)!+ . . . ? Since e^a = 1+a/1!+a^2/2!+...+a^(b-1)/(b-1)!+a^b/b!+a^(b+1)/(b+1)! + ..., we can write a^b/b!+a^(b+1)/(b+1)! + ... = e^a - (1+a/1!+a^2/2!+...+a^(b-1)/(b-1)!). This is a closed form for the series on the left. If b were large, this might not be much help. The question as to whether there is a special function giving the value of a^b/b!+a^(b+1)/(b+1)! + ... remains. We worked on this question for a while, trying various things and looking in a book of special functions. We found nothing, but an expert in special functions might be able to answer your question. -Doctor Jerry, The Math Forum Check out our web site! http://mathforum.org/dr.math/
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