2. Arcsinx = x+1/6x3+o (x 3), which is the arcsine expansion of Taylor's formula. Arcsinx can be replaced by Taylor formula expansion when seeking the limit.
3.tanx = x+ 1/3x 3+O (x 3), which is the tangent expansion of Taylor's formula. Tanx can be replaced by Taylor formula expansion when seeking the limit.
4.arctanx = x- 1/3x 3+O (x 3), which is the arctangent expansion of Taylor formula. When seeking the limit, we can use Taylor formula expansion instead of Arctanx.
5. ln (1+x) = x-1/2x2+o (x2), which is the expansion of Taylor's formula. When seeking the limit, ln( 1+x) can be replaced by Taylor formula expansion.
6. Cosx =1-1/2x2+O (x2), which is the cosine expansion of Taylor's formula. When calculating the limit, Taylor formula expansion can be used instead of Cosx.
Extended data:
Taylor theorem initiated the finite difference theory, so that any univariate function can be expanded into a power series; Meanwhile, Taylor became the founder of finite difference theory. Taylor also discussed the application of calculus in a series of physical problems, among which the result of transverse vibration of strings is particularly important.
He deduced the basic frequency formula by solving the equation, which initiated the study of string vibration. In addition, this book also includes his other creative work in mathematics, such as discussing singular solutions of ordinary differential equations and studying curvature problems.
The importance of Taylor's unfolding is reflected in the following five aspects:
The derivation and integration of 1 and power series can be carried out item by item, so the summation function is easier.
2. The analytic function can be extended to the analytic function defined on the slice on the complex plane, and the complex analysis method is feasible.
3. Taylor series can be used to approximate the function value and estimate the error.
4. Prove the inequality.
5. Find the limit of the formula to be determined.
References:
Baidu Encyclopedia-Taylor Unfolding