incorrect diffrentiation

Thread Starter

recca02

Joined Apr 2, 2007
1,214
this is an old trick at least for the biggies here but please care to elaborate on the below example

1+1+1......n times=n
1+1+1... x times = x ...x is a variable -----(1)
x+x+x....n times = x.n
x+x+x....x times = x^2
diff w.r.t x
1+1+1... x = 2.x
from (1)
x = 2.x
or
1 = 2
i know diff isnt allowed since limit does not exist in the case of
x+x+x....x times
but somehow i want a little detailed explanation
wud highly appreciate it if someone can help.
 

Thread Starter

recca02

Joined Apr 2, 2007
1,214
but d/dx(x^2) = is 2.x (that is what i intended)
i mean diff both rhs and lhs
there is nothin wrong with diff the only thing is it is not allowed in this case.
 

DrNick

Joined Dec 13, 2006
110
I think your problem is that you are using a dummy variable as part of the limit in your sum, making the summation non-linear. So you have
sum(x, i=0, x)

if we differentiate both sides you will notice that since the summation is no longer linear we cant interchange the order of summation and differentiation. SO,

d/dx ( sum(x, i=0, x) ) = d/dx(x^2) != [ sum(d/dx(x), i=0, x) = sum(1, i=0, x) ]

that x that is the upper limit must also be considered when differentiating (this is why we can't bring the derivative inside of the sum).
 

Dave

Joined Nov 17, 2003
6,970
but d/dx(x^2) = is 2.x (that is what i intended)
i mean diff both rhs and lhs
there is nothin wrong with diff the only thing is it is not allowed in this case.
Apologies I misinterpreted your initial question. I concur with the answer offered by DrNick.

Dave
 

Thread Starter

recca02

Joined Apr 2, 2007
1,214
thanks mr. dave and drnick
anyway can anyone come up with the graph of such function
i mean x+x+x+.....x times
wud it be a step funtion with a linear component
 

Dave

Joined Nov 17, 2003
6,970
thanks mr. dave and drnick
anyway can anyone come up with the graph of such function
i mean x+x+x+.....x times
wud it be a step funtion with a linear component
Plotting (sum of x) against (i = 1 to x) gives a straight line of gradient x. Try it in Excel with x arbitrarily set to say 10.

Dave
 

Thread Starter

recca02

Joined Apr 2, 2007
1,214
i'll bother u just once more.
how much sense does it make that a variable x which is a real number lets say
1.786 to be added 1.786 times ( X+X+..X times)
its been years since i learned summation but if i m not wrong it adds only
whole numbers it does not take into account fractional values.
 

Dave

Joined Nov 17, 2003
6,970
i'll bother u just once more.
how much sense does it make that a variable x which is a real number lets say
1.786 to be added 1.786 times ( X+X+..X times)
its been years since i learned summation but if i m not wrong it adds only
whole numbers it does not take into account fractional values.
Although it sounds wierd, 1.786 to be added 1.786 times is the same as 1.786^2 - think of it in terms of integers for a proof, and think that the rules apply equally to all real numbers.

Dave
 
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