An exponential series is a sum of terms where each term is a constant multiple of the previous term, raised to an increasing power. For example, ex=1+x+x22!+x33!+…e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dotsex=1+x+2!x2+3!x3+….
The series expansion of exe^xex around x=0x = 0x=0 is 1+x+x22!+x33!+…1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots1+x+2!x2+3!x3+….
A logarithmic series involves terms where each subsequent term is derived from the previous one by multiplying by a decreasing power of a variable, typically logarithmic in nature.
Example of Logarithmic Series
The series expansion of ln(1+x)\ln(1+x)ln(1+x) around x=0x = 0x=0 is x−x22+x33−…x - \frac{x^2}{2} + \frac{x^3}{3} - \dotsx−2x2+3x3−….
Exponential and logarithmic series are fundamental in calculus, differential equations, and various physical models like population growth, radioactive decay, and signal processing.
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