Argumenting for this inequality involving the residual of a taylor series of ln

[lo2]

Homework Statement

OK I have to argument for the fact that this inequality is true, where x > 1.

[itex] |R_n \ln{x}| \leq \frac{1}{n+1}(x-1)^{n+1}[/itex]

And I have found out that the residual is equal to this:

[itex]R_n \ln{x} = \frac{1}{n!} \int^x_a{f^{n+1}(t)(x-t)^{n}dt}[/itex]

Homework Equations

The Attempt at a Solution

So since I have to just argument, not prove.

Can I just look at the parts [itex]\frac{1}{n!}[/itex] and [itex](x-1)^{n+1}[/itex], and then say that the first one goes towards zero very fastly as n increases whereas the other one goes towards infinity quite rapidly as well. And therefore this inequality must be true.

Also I perhaps should add that this part:

[itex] \frac{(x-1)^{n+1}}{n+1}[/itex]

Goes toward infinity as the exponential function out-sprints the linear function. Of course only for x > 2.

 

[mighty2000]

Thank you for sharing your thoughts on this inequality. I would like to contribute to this discussion by providing some additional arguments for the truth of this inequality.

Firstly, we can consider the term (x-1)^{n+1}. As x > 1, this term will always be positive. As n increases, this term will grow at a faster rate than the term 1/(n+1). This can be seen by considering the limit of (x-1)^{n+1}/(n+1) as n approaches infinity, which is equal to infinity. Therefore, this term will always dominate over the term 1/(n+1) and contribute to the overall inequality being true.

Secondly, let's look at the term R_n \ln{x}. This term represents the residual of the Taylor series for the natural logarithm function, evaluated at x. As n increases, the Taylor series becomes a more accurate approximation of the natural logarithm function. This means that the residual R_n \ln{x} will decrease in magnitude, making the overall inequality more likely to be true.

Furthermore, as x > 1, the natural logarithm function is an increasing function. This means that the value of R_n \ln{x} will also increase as x increases. As a result, the magnitude of R_n \ln{x} will be larger for larger values of x, making the overall inequality more likely to be true.

In conclusion, the terms (x-1)^{n+1} and R_n \ln{x} both contribute to the overall inequality being true, and their behavior as n and x vary support the truth of the inequality. I hope this helps to further your understanding of why this inequality holds true.