I observed the range of values to be ##-1 < x < 5##. Thus, the radius of convergence is 3. The next question in this assignment asks us to find a property of the function that explains why the radius is what it is. I see no relationship between the two however.
Okay, spoke to my professor. The limit of the ratio test evaluates out to ##\frac{1}{3} |x-2|##, which I essentially had in the previous post. We note that this value is not dependent on ##k##, so that when k goes to infinity the limit stays as ##\frac{1}{3} |x-2|##. The range of x values that...
Okay, so I incorrectly defined ##a_n## and ##a_{n+1}##; as a result, I misapplied the Ratio Test.
##a_{n+1}## should be ##\frac{1}{9} \frac{1}{k+1} (4-3k) c_{k} (x-2)^{k+1}##
In reapplying the Ratio Test, I observe the following:
##lim_{k \rightarrow \infty} |\frac {\frac{1}{9} \frac{1}{k+1}...
I'm quite confused.. Here is what I get for the first six terms in the Taylor series:
Term 0: ##(1+(2))^{4/3}##
Term 1: ##4/3 (1+2)^{1/3}(x-2)##
Term 2: ##\frac{(4/9) (1+2)^{-2/3}}{2!} (x-2)^2##
Term 3: ##\frac{-(8/27) (1+2)^{-5/3}}{3!} (x-2)^3##
Term 4: ##\frac{(40/81) (1+2)^{-8/3}}{4!}...
So I need an additional factor of (1/3)? I have the formula containing a 3-1, which is 1/3. Also, ##c_k## is the coefficient of the kth term in the series (i.e. the number before the ##x-2##), therefore I do not think it needs a factor of ##x-2##.
Note, the coefficients of the series are:
f(x)...
Homework Statement
Let f(x)= (1+x)4/3 - In this question we are studying the Taylor series for f(x) about x=2.
This assignment begins by having us find the first 6 terms in this Taylor series. For time, I will omit them; however, let's note that as we continuously take the derivative of this...
Homework Statement
What is the diameter of a copper sphere that has the same mass as a 9.00 cm× 9.00 cm× 9.00 cm cube of aluminum?
Density of Aluminum = P(al) = 2.70g/cm3
Density of Copper = P(cu) = 8.96 g/cm3
Volume of Aluminum Cube = Vcube = 729 cm3
Homework Equations
Volume of a Sphere =...