1. ## Lagrange Error/Taylor's Theorum

I have a final exam coming up, and it has been a while since I've done problems with the Lagrange form of the remainder.

R = f^(n+1) (c)/(n+1)! * (x - a)^ (n+1) where f^(n+1) (c) is a maximum value.

Does anyone have any helpful general tips on finding c? I know it's a vague question, so I'll try to clarify. For example, I know whatever number derivative of a function like e^x will always be e^x, so finding a maximum value for that is easy, just plug in e^(bounding value), as is a trig function like sin or cosine, which is bounded by 1. But I start getting confused when the problem doesn't have a clear maximum point, and I don't know what to use for c.

For instance, here's a question from a past exam.

The series is sum (from 1 to infinity) of (-1)^n ln(n)/n^p

The first 2 parts of the question ask for what values of p is the series convergent and what for what values is it absolutely convergent? I'm pretty sure the answer is for p>1 in both cases but could someone tell me if I'm right? ^^;

The third part of the question is where I'm a bit confused. It says:

"Consider the case when p=2. How many terms of the series do we need to add in order to estimate the sum with accuracy |Error|< 0.01. (Note: use the inequality ln(n)<=n. )"

I'm assuming this means use n for the maximum term f^(n+1) (c), so that the equation becomes (n)/(n+1)! * (x - a)^ (n+1). But I'm not entirely sure what to do from there in terms of what to plug in.

Also, there's this example from my notes last year that sort of explains what my question is.

"Determine the order of the Taylor polynomial about x = 1 that can approximate ln(1.2) so that the error is less than 0.001."

I know that R = f^(n+1)(c)/(n+1)! * (x-1)^(n+1), but the hard part is finding what to put for the general derivative term f^(n+1) (c). My notes have an expression there, (-1)^(n+1) * n!/x^(n+1), which I know is a general term for the derivative of ln. But this isn't obvious, and I'm not sure I could have figured it out right away.

So all this basically comes down to how to figure out general terms for derivatives of functions...are there common ones I should know, or is this all based on intuition?

(By the way, sorry about the typed form of the equations, I'm not sure how to use the math script)

2. All what you know about c is that it is one some place between x and a, so depending on where you are looking at, which side of "a" you're standing, and how for of a you are you will have a different c.

$f(x)=f(a)+f'(a)(x-a)+\displaystyle\frac{f^2(a)(x-a)^2}{2!}+...+\displaystyle\frac{f^n(a)(x-a)^n}{n!}+R_n(x)$
Where $R_n(x)$ is the reminder.
And: $R_n(x)=\displaystyle\frac{f^{n+1}(c)(x-a)^{n+1}}{(n+1)!}$
And what you know about c is that $a

The series is sum (from 1 to infinity) of (-1)^n ln(n)/n^p

The first 2 parts of the question ask for what values of p is the series convergent and what for what values is it absolutely convergent? I'm pretty sure the answer is for p>1 in both cases but could someone tell me if I'm right? ^^;