Current location - Training Enrollment Network - Mathematics courses - Crazy calculus of happy mathematics
Crazy calculus of happy mathematics
35: Coefficient of (x- 1) 5 in Taylor expansion of function xlnx:

f(x)=xlnx,

f( 1)=0

f'=lnx+ 1

f''= 1/x

f'''=- 1/x?

f^(4)=2! /x?

f^(5)=-3! /x^4

f^(k)=(- 1)^k.(k-2)! /x^(k- 1)

(x- 1) 5 coefficient f (5) (1)/5! =-3! /5! =- 1/20

Choose a.

36: Series σ (n = 1, ∞) x n/2 n n n/n! Convergence radius of.

If x=2, it becomes σ (n = 1, ∞) n n/n! , divergent;

Sterling formula uses factorial n! ≈√(2nπ)(n/e)^n

σ(n= 1,∞)x^n/2^n.n^n/n!

≈σ(n= 1,∞)x^n/2^n.n^n/√(2nπ)(n/e)^n

=σ(n= 1,∞)(x/2)^n)./√(2nπ)/e^n

=σ(n= 1,∞)(ex/2)^n)./√(2nπ)

=( 1/√(2π))σ(n= 1,∞)(ex/2)^n)/n

(ex/2)^n<; 1,ex/2 & lt; 1,x & lt2/e,

Convergence radius 2/e

Choose c;

37: Use approximate formula:

sinx=x-x? /3! , | x |< 1 (radian), what is the numerical error?

Use the remainder formula.

rn=f^(n+ 1)(θ)(x-x0)^(n+ 1)/(n+ 1)!

The fourth term of the series is 0, there is no error, and the remainder of R4 is considered.

R4=f^(5)(θ)x^5/5!

f=sinx,f'=cosx,f''=-sinx,f'''=-cosx,f^(4)=sinx,f^(5)=cosx

R4=cos(θ)x^5/5! & ltcos( 1)/ 120=0.0045

Choose C.