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प्रश्न
Write down the Taylor series expansion, of the function log x about x = 1 upto three non-zero terms for x > 0
उत्तर
Let f(x) = log x
Taylor series of f(x) is
f(x) = `sum_("n" = 0)^oo "a"_"n" (x - 1)^"n"`
Where an = `("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
fIV(x) = `- 6/x^4`, fIV(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 + ...`
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