Q2Mathematics I

Question

Write the power series expansion of logarithm function $lo g (1 + x)$ .

Answer 1

Using the fundamental Taylor (or Maclaurin) series expansion centered exactly at $x = 0$ , the logarithmic function $f (x) = lo g (1 + x)$ can be expanded into an infinite alternating polynomial series.

lo g (1 + x) = x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - \frac{x ^{4}}{4} + \frac{x ^{5}}{5} - \dots

This specific mathematical series is absolutely crucial in calculus and numerical analysis. It is strictly valid and mathematically converges only when the variable $x$ falls within the precise interval $(- 1, 1]$ .

Q2Mathematics I

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