Consider the function f(x) = 10^x
a) write the power series for f(x) centered at a = 2 using summation notation
b) Suppose you wanted to estimate the function "f" at values of x in the interval [1.9,2.1] using the 4th degree Taylor polynomial T(4)(x) for f(x) centered at a = 2. Use Taylor's inequality to estimate the least upper bound for |R(4)(x)|.
Note: answer both a & b since they correspond with eachother
a) To find the power series for the function f(x) = 10^x centered at a = 2 using summation notation, we can use the Taylor series expansion for the exponential function e^x. Since 10^x = e^(x ln(10)), we have:
f(x) = e^(x ln(10))
The Taylor series expansion for e^x centered at a = 0 is given by:
e^x = 1 + x + (x^2) / 2! + (x^3) / 3! + (x^4) / 4! + ...
To shift the series to the right by 2 units, we replace x with (x - 2):
e^(x - 2) = 1 + (x - 2) + ((x - 2)^2) / 2! + ((x - 2)^3) / 3! + ((x - 2)^4) / 4! + ...
Now, substituting x ln(10) in place of x, we get:
f(x) = e^((x - 2) ln(10)) = 1 + (x - 2) ln(10) + ((x - 2)^2 ln(10)^2) / 2! + ((x - 2)^3 ln(10)^3) / 3! + ((x - 2)^4 ln(10)^4) / 4! + ...
Therefore, the power series for f(x) centered at a = 2 using summation notation is:
f(x) = ∑ ((-1)^(n-1) (x - 2)^n ln(10)^n) / n!, where n ranges from 1 to ∞.
b) To estimate the least upper bound for |R(4)(x)| using Taylor's inequality, we need to know the remainder term R(n)(x) for the Taylor polynomial T(n)(x) of the function f(x) centered at a = 2.
The remainder term R(n)(x) is given by the formula:
R(n)(x) = (f^(n+1)(c))(x - a)^(n+1) / (n+1)!
where c lies between a and x.
In this case, the 4th degree Taylor polynomial T(4)(x) of f(x) centered at a = 2 is:
T(4)(x) = f(2) + f'(2)(x - 2) + f''(2)((x - 2)^2) / 2! + f'''(2)((x - 2)^3) / 3! + f''''(2)((x - 2)^4) / 4!
To estimate the least upper bound for |R(4)(x)|, we need to find the maximum value of |f''''(c)| on the interval [1.9, 2.1]. Here, f''''(c) represents the 5th derivative of f(x).
Taking the 5th derivative of f(x) with respect to x, we have:
f''''(x) = ln(10)^4 * 4!
Since ln(10) is a constant, the maximum value of |f''''(c)| is ln(10)^4 * 4!.
Using Taylor's inequality, the remainder term can be bounded by:
|R(4)(x)| <= (ln(10)^4 * 4!) * ((x - 2)^5) / 5!
Since we need to estimate the least upper bound for |R(4)(x)| on the interval [1.9, 2.1], we substitute the maximum value of x - 2, which is 0.1:
|R(4)(x)| <= (ln(10)^4 * 4!) * (0.1^5) / 5!
Simplifying the expression, we can find the estimated least upper bound for |R(4)(x)|.
In the usual manner, you get
10^x = 10^2 + 100/1! ln10(x-2) + 100/2! (ln10 (x-2))^2 + 100/3! (ln10 (x-2))^3 + ...
You can see this by noting that 10^x = e^(ln10)^x = e^(ln10 x)and using the normal power series for e^u
Now you can do the estimations