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 consider:
also:
Now the integrand in [1] is strictly greater than the integrand in
[2] 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
RonL
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.
RonL
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:
or:
RonL
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.