Show that ln(4+x) is the sum of its Taylor Series for -2 <= x <= 2.
To show that ln(4+x) is the sum of its Taylor series, we need to find the coefficients of the series and check if the series converges to ln(4+x) within the given interval -2 <= x <= 2.
The Taylor series expansion of ln(4+x) is represented as:
ln(4+x) = c₀ + c₁x + c₂x² + c₃x³ + ...
To find the coefficients c₀, c₁, c₂, c₃, etc., we can use the formula for the nth derivative of ln(4+x) and evaluate it at x = 0.
Step 1: Find the nth derivative of ln(4+x).
The nth derivative of ln(4+x) can be obtained by applying the chain rule repeatedly:
dⁿ/dxⁿ [ln(4+x)] = (-1)^(n-1) * (n-1)! / (4+x)^n
Step 2: Evaluate the nth derivative at x = 0.
Substituting x = 0 into the nth derivative formula, we get:
dⁿ/dxⁿ [ln(4+x)]|ₓ₌₀ = (-1)^(n-1) * (n-1)! / 4ⁿ
Note that when n = 0, the derivative evaluates to ln(4).
Step 3: Determine the coefficients of the Taylor series.
The coefficients cₙ of the Taylor series are given by the formula:
cₙ = (1/n!) * dⁿ/dxⁿ [ln(4+x)]|ₓ₌₀
Substituting the expression for the nth derivative, we can write:
cₙ = (1/n!) * [(-1)^(n-1) * (n-1)! / 4^n]
Simplifying the expression, we get:
cₙ = (-1)^(n-1) / (n * 4^n)
We have found the coefficients of the Taylor series. Now let's check if the series converges to ln(4+x) within the given interval -2 <= x <= 2.
The Taylor series expansion for ln(4+x) will converge to ln(4+x) if and only if the series converges for the values of x within the interval -2 <= x <= 2.
The radius of convergence of ln(4+x) is determined by the formula:
R = Lim(n→∞) |cₙ / cₙ₊₁|
Using the expression for cₙ, we can calculate the limit:
Lim(n→∞) |cₙ / cₙ₊₁| = Lim(n→∞) |((-1)^(n-1) / (n * 4^n)) / ((-1)^n / ((n+1) * 4^(n+1)))|
Simplifying the expression, we get:
Lim(n→∞) |cₙ / cₙ₊₁| = Lim(n→∞) |-4(n+1) / n|
Taking the limit as n approaches infinity, we find:
Lim(n→∞) |-4(n+1) / n| = 4
Since the limit is less than infinity, the series has a radius of convergence of R = 4. Therefore, the series converges for all x within the interval -4 < x < 4.
Since -2 <= x <= 2, which is within the interval (-4, 4), we can conclude that the given interval is within the convergence range of the Taylor series expansion of ln(4+x).
Therefore, ln(4+x) is indeed the sum of its Taylor series for -2 <= x <= 2.