Alternating partial sums of a series

[spaghetti3451]

Consider the Taylor series expansion of $e^{-x}$ as follows:

$\displaystyle{e^{-x}=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{6}+\dots}$

For $x>0$, the partial sums $1$, $1-x$, $\displaystyle{1-x+\frac{x^{2}}{2}}$ bound $e^{-x}$ from above and from below alternately.

How do I prove this?

 

[haruspex]

Consider the Taylor series expansion of $e^{-x}$ as follows:

$\displaystyle{e^{-x}=1-x+\frac{x^{2}}{2}-\frac{x^{3}}{6}+\dots}$

For $x>0$, the partial sums $1$, $1-x$, $\displaystyle{1-x+\frac{x^{2}}{2}}$ bound $e^{-x}$ from above and from below alternately.

How do I prove this?

Be clear what you are trying to prove: that
1 > e^-x, and
1-x < e^-x, etc.

 

[spaghetti3451]

Yes.

I am looking for a proof that covers all the cases $1>e^{-x}, 1-x<e^{-x}, \dots$ at once.

 

[haruspex]

Yes.

I am looking for a proof that covers all the cases $1>e^{-x}, 1-x<e^{-x}. \dots$ at once.

Ok, so what does it say about the sum of the rest of the terms in the expansion in each case?

 

[spaghetti3451]

Well, it say that $1-e^{-x}>0, 1-x-e^{-x}<0, \dots$.

 

[haruspex]

I am looking for a proof that covers all the cases 1>e−x,1−x<e−x,…1>e−x,1−x<e−x,…1>e^{-x}, 1-x

Ok, that edit happened while I was replying.
That was not clear from the OP. It looked like you were only asking for those three specific partial expansions.
Anyway, my hint in post #4 still applies.

 

[spaghetti3451]

But, it's difficult to start the general proof since there are two separate cases.

Are we supposed to prove two inequalities, one for even partial sums and one for odd partial sums?

 

[mathwonk]

this is a typical alternating series. each term is smaller than the previous one and the signs alternate. It follows that any partial sum ending in a plus sign is larger than any partial sum ending in a minus sign, hence also larger than the full sum, etc...I.e. the partial sums are jumping back and forth across the full sum, or final limit. draw a picture, as this is essentially obvious.

i.e. start from home and go a block north. then go back a 1/2 block south. you are obviously somewhere between home and the first block. then go back 1/4 block north. you are obviously somewhere between the 1/2 block point and the one block point...

in answer to your question, yes there do seem to be two cases.the best discussion i know of for series including alternating ones is in courant's calculus.

here is a free copy:

https://archive.org/details/DifferentialIntegralCalculusVolI

 

[spaghetti3451]

The qualititative argument works well for me.

But, just for further info, do you think Taylor's theorem can be used to write a proof?