How to use Euler formula to expand the functions exp (x) cosx and exp (x) SiNx into power series of X?

How to use Euler formula to expand the functions exp (x) cosx and exp (x) SiNx into power series of X?

cosx=[e^ix+e^(-ix)]/2
e^x cosx=[e^(x+ix)+e^(x-ix)]/2
=1/2*∑[(x+ix)^n+(x-ix)^n]/n!
=1/2* ∑[x^n/n!*( (1+i)^n+(1-i)^n]
because
1+i=√2(cosπ/4+isinπ/4)
1-i=√2[cos(-π/4)+isin(-π/4)]
(1+i)^n+(1-i)^n=(√2)^n* 2cosnπ/4
So e ^ xcosx = ∑ [x ^ n / N! * (√ 2) ^ n Cosn π / 4]
Similarly:
sinx=[e^ix-e^(-ix)]/2i
e^x sinx=[e^(x+ix)-e^(x-ix)]/2i
=1/2*∑[(x+ix)^n-(x-ix)^n]/n!
=1/2* ∑[x^n/n!*( (1+i)^n+(1-i)^n]
because
1+i=√2(cosπ/4+isinπ/4)
1-i=√2[cos(-π/4)+isin(-π/4)]
(1+i)^n-(1-i)^n=(√2)^n* 2isin(nπ/4)
So e ^ xsinx = ∑ [x ^ n / N! * (√ 2) ^ n Sinn π / 4]

The function f (x) = SiNx is expanded into a power series of (x - π / 4) Do this: F (x) = SiNx = sin (x - π / 4 + π / 4) = root 2 / 2 (sin (x - π / 4) + cos (x - π / 4)) Let's talk about the direct substitution of X - π / 4 into the McLaughlin expansion of SiN x and COS x, OK?

Yes, because the McLaughlin formulas of SiNx and cosx hold true for all real numbers

SiNx = (e ^ x-e ^-x) / 2 is expanded into a power series of X

You mean SHX. Expand e ^ X and e ^ - x respectively and add them
e^x=1+x+x^2/2!+ x^3/3!+...+ x^n/n!+...
e^-x=1-x+x^2/2!- x^3/3!+...+ (-1)^nx^n/n!+...
shx=(e^x-e^-x)/2=x+x^3/3!+ x^5/5!+...+ x^(2n-1)/(2n-1)!+...

Help do some microeconomic problems Suppose that manufacturer AB and manufacturer B are competitors producing the same and different products in a market, the demand curve for manufacturer a is pa = 200-qa, and the demand curve for manufacturer B is Pb = 300-0.5qb. The current sales volumes of the two manufacturers are QA = 50 and QB = 100 respectively; (1) What are the price elasticity of demand EDA and EDB of manufacturers a and B? (2) If the price reduction of manufacturer B increases the demand of manufacturer B to 160 and reduces the demand of manufacturer a to 40, what is the cross price elasticity EAB of the demand of manufacturer a? (3) If B pursues the maximization of sales revenue, do you think manufacturer B's price reduction is a correct behavior choice?

1、edA=-3 edB=-5
2、11/6
3. Correct behavior

Microeconomic issues Q = 0.02m-2p, M = 7500, P = 30 yuan. Q. when the price rises to 40 yuan, what is the price effect of the price rise? How many bottles are the substitution effects? What is the income effect?

When p = 30 yuan, q = 0.02m-2p = 150-60 = 90 bottles
P = 40 yuan,
M = 7500, q = 0.02m-2p = 150-80 = 70 bottles, the price effect of price rise is 90-70 = 20 bottles
They are all substitution effects. The income has not changed, and the income effect is 0

Derivative problem f(x)=x^3/(x+1)

y=x^3/(x+1),
y'=(x^3)'/(x+1)+x^3 × [1/(x+1)]‘
=3x ²/ (x+1)+x^3[-1/(x+1) ²]= (2x^3+3x ²)/ (x+1) ²