Maclaurin functions. Expansion of functions into power series
If the function f(x) has derivatives of all orders on a certain interval containing point a, then the Taylor formula can be applied to it:
,
Where r n– the so-called remainder term or remainder of the series, it can be estimated using the Lagrange formula:
, where the number x is between x and a.
f(x)=
at point x 0 = Number of row elements 3 4 5 6 7
Use decomposition elementary functions e x , cos(x), sin(x), ln(1+x), (1+x) m
Rules for entering functions:
If for some value X r n→0 at n→∞, then in the limit the Taylor formula becomes convergent for this value Taylor series:
,
Thus, the function f(x) can be expanded into a Taylor series at the point x under consideration if:
1) it has derivatives of all orders;
2) the constructed series converges at this point.
When a = 0 we obtain a series called the Maclaurin series:
,
Expansion of the simplest (elementary) functions in the Maclaurin series:
Exponential functions
, R=∞
Trigonometric functions 
, R=∞ 
, R=∞
, (-π/2< x < π/2), R=π/2
The function actgx does not expand in powers of x, because ctg0=∞
Hyperbolic functions
Logarithmic functions
, -1
