Question

Найти первые пять членов ряда Тейлора для функции y= ln(10+x) в окрестности точки Xo = 0. Помогите пожалуйста!

Answer 1

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Answer 2

Finding the Taylor Series for the Function y = ln(10+x) at x = 0

To find the Taylor series for the function y = ln(10+x) in the neighborhood of the point x₀ = 0, we can use the formula for the Taylor series expansion:

Taylor Series Formula:

The Taylor series expansion of a function f(x) centered at x = a is given by:

f(x) = f(a) + f'(a)(x - a)/1! + f''(a)(x - a)²/2! + f'''(a)(x - a)³/3! + ...

where f'(a), f''(a), f'''(a), etc., represent the derivatives of the function evaluated at x = a.

In this case, we want to find the Taylor series expansion of y = ln(10+x) centered at x₀ = 0.

Step 1: Find the derivatives of y = ln(10+x)

Let's start by finding the first few derivatives of y = ln(10+x):

y' = 1/(10+x) y'' = -1/(10+x)² y''' = 2/(10+x)³ y'''' = -6/(10+x)⁴

Step 2: Evaluate the derivatives at x = 0

Now, let's evaluate these derivatives at x = 0:

y'(0) = 1/10 = 0.1 y''(0) = -1/100 = -0.01 y'''(0) = 2/1000 = 0.002 y''''(0) = -6/10000 = -0.0006

Step 3: Write the Taylor series expansion

Using the Taylor series formula, we can write the Taylor series expansion for y = ln(10+x) centered at x₀ = 0:

y(x) = y(0) + y'(0)(x - 0)/1! + y''(0)(x - 0)²/2! + y'''(0)(x - 0)³/3! + ...

Substituting the values we found earlier:

y(x) = 0 + 0.1x + (-0.01)(x²)/2 + 0.002(x³)/6 + ...

Simplifying the terms:

y(x) = 0.1x - 0.005x² + 0.000333x³ + ...

Therefore, the first five terms of the Taylor series for the function y = ln(10+x) in the neighborhood of x₀ = 0 are:

y(x) = 0.1x - 0.005x² + 0.000333x³ - ...

Please note that the Taylor series expansion is an infinite series, and we have only provided the first few terms. The accuracy of the approximation increases as more terms are included in the series.