Problem involving Taylors remainder theorem

Fzang · 2010-10-07 04:00:53

Okay, so I have this

content
for x > 1

In English the text says "Explain, from Taylors remainder theorem that *above statement is true (?)* for x > 1"

Tell me if it didn't quite made sense. I'm having some problems translating math language.

Can anyone get me started? I have no idea where I'm supposed to start at, or what I'm doing.

bobbym · 2010-10-07 11:10:53

Hi Fzang;

The notation is a little muddy.

All that is saying up there is that the Absolute value of the difference between the function ln(x) and its Taylor approximation is less than or equal to the term on the far right.

Do you need a derivation of that error term?

Fzang · 2010-10-07 19:05:18

Yes. I think that's what the problem asks of me.

bobbym · 2010-10-07 19:21:13

The Taylor expansion of ln(x) at 1 is:

We notice we have an alternating series. The tail of an alternating series is always less than the first neglected term.
So:

The remainder Rn is equal to

Now we know theoretically that

Where Tn is the truncated Taylor series.

Since Rn is an alternating series it is less than the first neglected term ( leaving out the absolute value here), so

Which is what we wanted to show.

Fzang · 2010-10-07 19:22:04

I think it needs fixing

Edit: you fixed it before I could post.

Last edited by Fzang (2010-10-07 19:22:20)

bobbym · 2010-10-07 19:59:00

Hi;

The meat is right up there. It may need a little cleaning but it is okay.
If your teacher knows anything about alternating series and the first term ( due to Leibnitz I think ) then he/she will find it okay.
Also I will be adding some refinements as I think of them.

George,Y · 2010-10-09 00:51:48

Great Post

Fzang · 2010-10-10 09:55:50

The tail of an alternating series is always less than the first neglected term

Could you elaborate please? I'm not sure what you're talking about. Which part of the series is the 'tail'?

Last edited by Fzang (2010-10-10 10:01:56)

bobbym · 2010-10-10 10:28:05

The tail is the terms left over.

Consider the problem of trying to determine this sum by computer.

If you sum this well known alternating harmonic series to nine terms you get:

In numerical analysis we are concerned with the error estimate.
My brother says that numerical analysts are the exact opposite of chaoticians.
While chaoticians are trying to magnify the error to study it we are trying to eliminate it or at least estimate it.
In numerical analysis a number without an error estimate, is meaningless!

So the tail is:

We would like to estimate what the tail is or bound it.

We want to know how close we are to:

If we only sum manually the first nine terms as we have done in 1)

We can do that by estimating the tail ( 2 ) ! Do you follow so far?

Math Is Fun Forum

#1 2010-10-07 04:00:53

Problem involving Taylors remainder theorem

#2 2010-10-07 11:10:53

Re: Problem involving Taylors remainder theorem

#3 2010-10-07 19:05:18

Re: Problem involving Taylors remainder theorem

#4 2010-10-07 19:21:13

Re: Problem involving Taylors remainder theorem

#5 2010-10-07 19:22:04

Re: Problem involving Taylors remainder theorem

#6 2010-10-07 19:59:00

Re: Problem involving Taylors remainder theorem

#7 2010-10-09 00:51:48

Re: Problem involving Taylors remainder theorem

#8 2010-10-10 09:55:50

Re: Problem involving Taylors remainder theorem

#9 2010-10-10 10:28:05

Re: Problem involving Taylors remainder theorem

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