Question

Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0

Answer 1

Let f(x) = log x

Taylor series of f(x) is

f(x) = `sum_("n" = 0)^oo "a"_"n" (x - 1)^"n"`

Where a_n = `("f"^"n"(1))/("n"!)`

f(x) = log x, f(1) = log 1 = 0

f'(x) = `1/x`, f'(1) = 1

f''(x) = `- 1/x^2`, f"(1) = – 1

f'''(x) = `2/x^3`, f'''(1) = 2

f^IV(x) = `- 6/x^4`, f^IV(1) = – 6

∴ The required expansion of the function is log x

= `0 + 1 ((x - 1))/(1!) - 1(x - 1)^2/(2!) + 2(x - 1)^3/(3!) - 6(x - 1)^4/(4!) + ...`

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