My math prof is giving us an opportunity to earn back some points on our midterms, so I cam to ask for a spot of help on some problems that I missed some stuff on, hopefully you guys can give me a hand:
#1) Finding the exact value of this series:
The sum from k = 0 to infinity of (2^(2k+1))/(5^(k))
When I did this out, I think that I said it added up to something, as after the first couple terms, they began to be less than one, but I guess that I was mistaken as to the actual value...any thoughts?
#2) Compute the MacLaurin Series for e^x to compute the following:
Indefinite Integral of (e^(x^2))/(x) as an infinite series.
Here, I didn't really get anywhere, and he cautioned the class that you had to actually split off a term, leaving the index for the series representation to be 1 instead of zero. I wasn't exactly sure as to how we're supposed to go about doing that...
#3) Find the 2nd Taylor polynomial, T2 (x), for f (x) = tan x centered at a= 0. What’s the maximum error in this approximation if 0 ≤ x ≤ π 6 ? (Hint: both sec x and tan x are increasing functions on the given interval.)
For this one, I got pretty far, only missing the last step or so, but I guess that it was off by a bit, and I was wondering if someone could walk through the steps so that I could see where I went wrong with this. (I think I had all of the derivatives and consequent values right, but not sure about the rest)
Thanks in advance.