Let be given by
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 ?
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?
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