Use the binomial series to find the Taylor series about 0 for the function
f(x) = (1 + x)−3/5,
giving all terms up to the one in x4.
I have the answer; f(x)=1-3/5(x) +12/25(x^2) -52/125(x^3) + 234/625(x^4)
but i am not sure how to work out the interval validity and Use this series and the Taylor series for sinx to find the quartic Taylor polynomial about 0 for the function
f(x) = (sinx)/(1+x)^3/5
To find the Taylor series expansion for the function f(x) = (1 + x)^(-3/5) using the binomial series, follow these steps:
Step 1: Determine the general form of the binomial series.
The binomial series is given by the formula:
(1 + x)^n = C(0,n) + C(1,n)x + C(2,n)x^2 + C(3,n)x^3 + ...
where C(r, n) represents the binomial coefficient, given by:
C(r, n) = n! / (r!(n-r)!)
Step 2: Identify the values of n and r.
In this case, we have f(x) = (1 + x)^(-3/5), so n = -3/5.
Step 3: Calculate the binomial coefficients.
Using the formula C(r, n) = n! / (r!(n-r)!), we can calculate the coefficients required for the expansion.
C(0, -3/5) = (-3/5)! / (0!((-3/5)-0)!)
C(1, -3/5) = (-3/5)! / (1!((-3/5)-1)!)
C(2, -3/5) = (-3/5)! / (2!((-3/5)-2)!)
C(3, -3/5) = (-3/5)! / (3!((-3/5)-3)!)
Simplify each coefficient using factorials, and replace negative factorials with their reciprocal positive values:
C(0, -3/5) = 1 / (1 * 2 * 3 * 4 * 5^(3/5))
C(1, -3/5) = -3/5 / (1 * 2 * 3 * 4 * 5^(3/5))
C(2, -3/5) = 12/25 / (1 * 2 * 3 * 5^(3/5))
C(3, -3/5) = -52/125 / (1 * 2 * 5^(3/5))
Step 4: Write out the expansion.
Using the calculated coefficients, the Taylor series expansion for f(x) = (1 + x)^(-3/5) is:
f(x) = 1 + (-3/5)x + (12/25)x^2 + (-52/125)x^3 + ...
To find the interval of validity, you need to consider the x-values for which the original function converges to the Taylor series. In this case, the Taylor series for f(x) is valid for x such that |x| < 1, since the binomial series only converges for values within this range.
To find the quartic Taylor polynomial for f(x) = (sinx) / (1+x)^3/5, you can multiply the Taylor series of sinx by the Taylor series for (1+x)^(-3/5) term by term, keeping terms up to x^4.
The Taylor series for sinx is: sinx = x - (x^3 / 3!) + (x^5 / 5!) - (x^7 / 7!) + ...
Multiply each term of the sinx series by the corresponding term of the (1+x)^(-3/5) series and collect terms up to x^4.