Consider the definite integral where n is a positive integer and lamda is a positive real number.
Given also that
Show that when and when
Help of any kind would be greatly appreciated.
Would the first part of the question have something to do with limits? We can see that and so the maximum value for the expression is obviously the limit as lamda approaches n since approaches infinite. However, im not sure how to evaluate the limit of the entire expression .
I'll attempt the second half.
Let
then
therefore,
Now
is a strictly decreasing function and n is an integer greater than or equal to 1, so it is maximum for n=1 or:
You could answer part 1 by assuming and following the same procedure, then show that D is increasing on the range and therefore is less than D at n = lambda.