Proving Inequalities in Taylor Series: A Scientific Approach

[transgalactic]

http://img131.imageshack.us/img131/1323/29081539pt7.th.gif

how am i supposed to prove that the function is bigger then its tailor series??

the taylor series are written in endless way.
its should be equal

how to prove that its bigger??

for the first i tried prove by induction:

the base case
<br /> f_1(x) = e^x - \left( 1 + x\right)\\<br />
i know that x>0
but it doesn't prove that
<br /> f_1(x)&gt;0<br />
??

 

[HallsofIvy]

You need to recognize that the problem is NOT to proved "the function is bigger then its tailor series". Both of these functions are "analytic" and so they are equal to their Taylor series. You are asked to show that the functions are greater than the Taylor polynomial up to power n, in the first case, and to second power for the second. The right sides are finite sums.

Since each function is equal to its Taylor series, the only difference is that the Taylor polynomial, of order n, is missing all terms past n.

If x> 0 all those missing terms are positive. (In the middle of your post you say "I know that x>0". HOW do you know that? Is it a condition of the problem? You don't mention that!)

 

[transgalactic]

you are correct i wasnt given that x>0
so my general function are bigger then their taylor series by the remainder of Rn

this is the logic
how to formulate it into equation??

 

[transgalactic]

how to prove it?

 

[transgalactic]

how to prove it?