lim e^(-x)* sum (-1)^n*binom(n,k)*(e^x-sum x^m/m!, m = 0...n),n = k...infinity if x tends to infinity?
The corrected solution of the problem: Since the following equality holds,
ex - 1 - x - ... - xn/n! = integral0x ex-u * un du / n! = ex * Gamma(n+1,x)/n! , for all x,
by changing the order of summation and integration, the sum of the series equals
(-1)k/k! * integral0x uk * e-2u du =(-1)k Gamma(k+1, 2x) / k! * 2k+1
which tends to (-1)k / 2k+1 , as x tends to infinity. The change of order of integration and summation is correct, since the functional series are uniformly convergent in every finite interval.
PS1. (an illustrative example) The series \sum0\infty (-1)n xn = 1/(1+x) tends to 1/2 as x tends to 1- even though the limits of the terms (-1)n do not form a convergent series. Indeed, xn is not covergent uniformly to the function equal 0 for x < 1 and equal 1 for x=1. However, the convergence is uniform in every interval of the form [0, a] with 0
One has a kind of confirmation by considering the sum of the series:
sumk=0\infty THE_SERIES(k,x) = sumn=0\infty sumk=0n (-1)n {binom(n,k)} \int0x e-uundu/n! =
sumn=0\infty (-2)n integral0x e-uundu/n! goes to 1/3, which is the same as sumk=0\infty (-1)k/ 2k+1.
THANKS
The author thanked the professors Ted Eisenberg, Ovidiu Furdui Technical University of Cluj-Napoca, Cluj-Napoca, Romania (the author of the problem 5456) , Peter Breuer and Joachim Domsta · for their support. I corrected my error and sent as an attachment the solution to problem 5456, SSMA. The error is entirely mine, it's not on the mathematical site http://wolframalpha.com. Your sincerely, Anna Tomova
Dear Anna Tomova,
Let me comment your current version of the enclosed paper: It is not a full solution, since it does not present the final formula for the limit of function F(x,k) defined in the problem by Ovidiu Furdui, as x goes to infinity.
Can you please answer why you are omitting the fact, that the limit equals (-1)k/2k+1 as x tends to infinity?
Note, that for this fact the uniform covergence of the series in the whole domain of all real x is not needed!
Best regards, Joachim Domsta
PS. I would appreciate your kind acknowledgements only in case, if you presented the final answer of the Ovidiu Furdui question: "What is the limit?", JoD
Dear prof. Joachim Domsta,
I can not understand why you do not accept my solution.The formula for the limit in my solution of 5456 is presented and proven quite correctly. I think that the derivation member by member requires one uniformly convergence, but the derivation is a locally concept, You are right. I'm very tankful You about the remark: "The change of order of integration and summation is correct, since the functional series are uniformly convergent in every finite interval." For me it was very curious not only to get the formula for this limit, but also the fact that the other relimit does not exist. The problem of prof. Ovidiu Furdui is very interesting - it shows how fine the mathematical objects are and we have to be careful not to mistaken. I have not written the translations in detail, but I have checked everything.I show bellow the answer of prof. Ted Eisenberg of SSMA for the same my solution.
"Dear Anna,
Thanks for your corrected solution to 5456; it is now fine. All the best."
Please, answer me there are some misunderstandings.
Best regards, Anna Tomova
- Badges
- Science topic
- Similar topics
- Social Psychology
- Values