Question

Find a formula for the power series of f(x) = 5 ln(1 + x), -1 < x < 1 in the form ∑∾ₙ₌₁

Answer 1

The power series of f(x) = 5 ln(1 + x) is given by f(x) = 5(x - x^2/2 + x^3/3 - x^4/4 + ...)

Step-by-step explanation:

To find the power series of the function f(x) = 5 ln(1 + x), we can use the power series expansion of the natural logarithm function and substitute the function into the expansion. The power series expansion of ln(1 + x) is given by:

ln(1 + x) = x - x^2/2 + x^3/3 - x^4/4 + ...

Substituting 5 for x in the expansion, we get:

5 ln(1 + x) = 5(x - x^2/2 + x^3/3 - x^4/4 + ...)

So, the power series of f(x) = 5 ln(1 + x) is:

f(x) = 5(x - x^2/2 + x^3/3 - x^4/4 + ...)