prove by induction that for every natural n and every x>0
(e^x) > 1 + x/1! + (x^2)/2! + ... + (x^n)/n!
it says we can assume that we know basic properties of derivatives and integrals, but i just am not sure where to begin.
In duction follows a pattern:Originally Posted by cen0te
1. First we show that what we wish to prove for general
n actualy holds for the first n which is realevant to the
problem. In this case it's n=0, and we need to show that
for every x>0 that:
,which I presume you know is true.
2. Second we show that if what we have to prove is true
when n=k, then it is also true for n=k+1. So assume it
true for n=y, then we have
for all .
Now the integrand in  is strictly greater than the integrand in
 at every point over which the integral is taken. So
which can be rewritten from what we have found above as:
So rearranging and replacing y by x we have:
3. That is we have proven that if what we have to prove for all ,
is true for n=k, then it is true for n=k+1, also we have proven that
it is true for n=0, so by the principle of induction we have proven
it true for all
Ok, I think I understand your proof. I understand induction, it is calculus that I am horribly rusty on. I have not taken any in over 5 years, and I do not even remember the notation.
Many thanks for your help. As an aside, could the statement be reworked to support x<0?
The problem is that the instructions given with the problemOriginally Posted by cen0te
indicate that you are expected to use calculus in the proof.
In fact I was a bit worried about this problem when I first
saw it because induction is not the natural way to prove it.
If you know the series expansion for the
result is hovering around the obvious pile.
I don't know if this is what they are looking for but: suppose ,Originally Posted by cen0te
and let , then , so for all (the set of natural
numbers 0,1, .. or 1,2, .. depending on your preference for how
they are defined).
now substitute for to get:
Hmm, I actually thought of that, but for some reason I thought that was basically just cheating. I had assumed that it would have something to do with how e^x with x<0 would just keep approaching 0. However, I can't come up with anything that satisfies me, so I will go with your more experienced suggestion. Thanks again.