Need derivative wrt n, of summation from 1 to n

gradstudent1229 · #1 $Old$ April 30th, 2008, 02:31 PM

I have a function, say f(i,n). I want to prove that it has a unique maximum at some n*. Thus I need the second derivative *with respect to n* of

sum from i=1 to n of f(i,n).

I don't know how to deal with the n in the summation. Does anyone have any ideas?

Opalg · #2 $Old$ May 1st, 2008, 01:50 AM

Quote:

Originally Posted by gradstudent1229 $View Post$

I have a function, say f(i,n). I want to prove that it has a unique maximum at some n*. Thus I need the second derivative *with respect to n* of

sum from i=1 to n of f(i,n).

I don't know how to deal with the n in the summation. Does anyone have any ideas?

You can't differentiate with respect to a discrete variable. It may sometimes be possible to replace discrete variables by continuous ones, so that for example $\sum_{i=1}^nf(i,n)$ becomes $\int_1^x\!\!\!f(t,x)\,dt$ . If this has a maximum at say x*, then the nearest integer to x* might (with luck) provide the maximum you're looking for.