Taylor Series

Overview

Engineering reference data for Taylor Series in miscellaneous.

Key Formulas

Unit Conversion

$y = x \cdot k$

Multiply by conversion factor.

Linear Interpolation

$y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$

Estimate between two known points.

Percentage

$p = \frac{\text{part}}{\text{whole}} \times 100\%$

Part as fraction of whole.

Variables

Symbol	Description	Unit
$x$	Input value	—
$y$	Output value	—
$k$	Conversion factor	—

Additional Series Examples

The Taylor series expansions for the natural logarithm are also commonly used in engineering calculations. Here are two important forms:

For the natural logarithm around x = 1, valid for x > 0:

$$\ln(x) = 2 \left( \frac{x-1}{x+1} + \frac{1}{3} \left( \frac{x-1}{x+1} \right)^3 + \frac{1}{5} \left( \frac{x-1}{x+1} \right)^5 + \cdots \right)$$

For the natural logarithm of (1+x), valid for -1 < x ≤ 1:

$$\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \cdots$$

References

• Original Source