Let be given by

QUESTION

1. Write down .

2. Write down the n-th stage Taylor's expansion for f, giving a clear statement about the remainder .

3. For which do we have as ?

4. What does this say about the range of validity of the Maclaurin series for log(1+x) on

5. What is the range of validity of the Maclaurin series for log(1+x) on ?

DISCUSSION

1.

2. What does this even mean? I have this statement in my notes

Suppose and all exist and are continuous on [a,b] (using left and right derivatives at a and b) and exists on (a,b). Then such that

Am I supposed to set a to 0 and set b to x? That would give me this thing

for ? Or is this Maclaurin expansion?

Here

3. I have no idea how to do this bit but I knowthe answer is x 1. All I can think to do is

but why on earth does this tend to 0 as n tends to infinity, if x 1 ??

4. The Maclaurin series is valid for 0 < x 1.

5. I kind of understand this part. the Maclaurin expansion is which we know converges absolutely for (f(0) = 0) so it will converge absolutely for so has radius of convergence at least 1. The maclaurin expansion is valid on (-1, 1]. Not at -1 because then log stops being defined. Is this right?

Any help with this would be very appreciated