Taylor, Binomial, Trigonometric Functions, Miscellaneous, Power Series

Taylor Series for Functions of one Variable

where Rn, the remainder after n terms, is given by either of the following forms:
Lagrange’s form Rn = f(n)(x - a)nCauchy’s form Rn =

The value ξ, which may be different in the two forms, lies between a and x. The result holds if f(x) has continuous derivatives of order n at least.

If limn → ∞ Rn = 0, the infinite series obtained is called the Taylor series for f(x) about x = a. If a = 0 the series is often called a Maclaurin series. These series, often called power series, generally converge for all values of x in some interval called the interval of convergence and diverge for all x outside that interval.

Binomial Series

Secial cases are
(a + x)2 = a2 + 2ax + x2
(a + x)3 = a3 + 3a2x + 3ax2 + x3
(a + x)4 = a4 + 4a3x + 6a2x2 + 4ax3 + x4
(1 + x)- 1 = 1 - x + x2 - x3 + x4 - ...        - 1 < x < 1
(1 + x)- 2 = 1 - 2x + 3x2 - 4x3 + 5x4 - ....        - 1 < x < 1
(1 + x)- 3 = 1 - 3x + 6x2 - 10x3 + 15x4 - ....        - 1 < x < 1

Series if Exponential and Logarithmic Functions

- ∞ < x < ∞
- ∞ < x <∞

Reversion of Power Series

If
y = c1x + c2x2 + c3x3 + c4x4 + c5x5 + ....
then
x = C1y + C2y2 + C3y3 + C4y4 + C5y5 + ....
where
c1C2 = - c2;
c15C3 = 2c22 - c1c3;
c17C4 = 5c1c2c3 - 5c23 - c12c4;
c19C5 = 6c12c2c4 + 3c12c32 - c13c3 + 14c24 - 21c1c22c1;
c111C6 = 7c13c2c5 + 84c1c23c3 + 7c13c3c4 - 28c12c2c32 - c14c6 - 28c12c22c4 - 42c25

Taylor Series for Functions of Two Variables

f(x, y) = f(a, b) + (x - a).fx(a, b) + (y - b).fy(a, b) + (1/2!).{(x - a)2.fxx(a, b) + 2(x - a)(y - b).fxy(a, b) + (y - b)2.fyy(a, b)} + ....
where fx(a, b), fy(a, b) .... denote partial derivatives with respect to x, y, .... evaluated at x - a, y - b.

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