Answer:
i think is the last one
Step-by-step explanation:
Answer:
20/24
Step-by-step explanation:
let the variable be x and x-4
5/6 = x-4/x
cross multiply
5x = 6x - 24
24= x
thus, x is 24 and x-4 is 20
<u>20/24</u>
Answer:
![f(x)=\sum_{n=1}^{\infty}(-1)^{(n-1)}2^{n}\dfrac{x^n}{n}](https://tex.z-dn.net/?f=f%28x%29%3D%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%28-1%29%5E%7B%28n-1%29%7D2%5E%7Bn%7D%5Cdfrac%7Bx%5En%7D%7Bn%7D)
Step-by-step explanation:
The Maclaurin series of a function f(x) is the Taylor series of the function of the series around zero which is given by
![f(x)=f(0)+f^{\prime}(0)x+f^{\prime \prime}(0)\dfrac{x^2}{2!}+ ...+f^{(n)}(0)\dfrac{x^n}{n!}+...](https://tex.z-dn.net/?f=f%28x%29%3Df%280%29%2Bf%5E%7B%5Cprime%7D%280%29x%2Bf%5E%7B%5Cprime%20%5Cprime%7D%280%29%5Cdfrac%7Bx%5E2%7D%7B2%21%7D%2B%20...%2Bf%5E%7B%28n%29%7D%280%29%5Cdfrac%7Bx%5En%7D%7Bn%21%7D%2B...)
We first compute the n-th derivative of
, note that
![f^{\prime}(x)= 2 \cdot (1+2x)^{-1}\\f^{\prime \prime}(x)= 2^2\cdot (-1) \cdot (1+2x)^{-2}\\f^{\prime \prime}(x)= 2^3\cdot (-1)^2\cdot 2 \cdot (1+2x)^{-3}\\...\\\\f^{n}(x)= 2^n\cdot (-1)^{(n-1)}\cdot (n-1)! \cdot (1+2x)^{-n}\\](https://tex.z-dn.net/?f=f%5E%7B%5Cprime%7D%28x%29%3D%202%20%5Ccdot%20%281%2B2x%29%5E%7B-1%7D%5C%5Cf%5E%7B%5Cprime%20%5Cprime%7D%28x%29%3D%202%5E2%5Ccdot%20%28-1%29%20%5Ccdot%20%281%2B2x%29%5E%7B-2%7D%5C%5Cf%5E%7B%5Cprime%20%5Cprime%7D%28x%29%3D%202%5E3%5Ccdot%20%28-1%29%5E2%5Ccdot%202%20%5Ccdot%20%281%2B2x%29%5E%7B-3%7D%5C%5C...%5C%5C%5C%5Cf%5E%7Bn%7D%28x%29%3D%202%5En%5Ccdot%20%28-1%29%5E%7B%28n-1%29%7D%5Ccdot%20%28n-1%29%21%20%5Ccdot%20%281%2B2x%29%5E%7B-n%7D%5C%5C)
Now, if we compute the n-th derivative at 0 we get
![f(0)=\ln(1+2\cdot 0)=\ln(1)=0\\\\f^{\prime}(0)=2 \cdot 1 =2\\\\f^{(2)}(0)=2^{2}\cdot(-1)\\\\f^{(3)}(0)=2^{3}\cdot (-1)^2\cdot 2\\\\...\\\\f^{(n)}(0)=2^n\cdot(-1)^{(n-1)}\cdot (n-1)!](https://tex.z-dn.net/?f=f%280%29%3D%5Cln%281%2B2%5Ccdot%200%29%3D%5Cln%281%29%3D0%5C%5C%5C%5Cf%5E%7B%5Cprime%7D%280%29%3D2%20%5Ccdot%201%20%3D2%5C%5C%5C%5Cf%5E%7B%282%29%7D%280%29%3D2%5E%7B2%7D%5Ccdot%28-1%29%5C%5C%5C%5Cf%5E%7B%283%29%7D%280%29%3D2%5E%7B3%7D%5Ccdot%20%28-1%29%5E2%5Ccdot%202%5C%5C%5C%5C...%5C%5C%5C%5Cf%5E%7B%28n%29%7D%280%29%3D2%5En%5Ccdot%28-1%29%5E%7B%28n-1%29%7D%5Ccdot%20%28n-1%29%21)
and so the Maclaurin series for f(x)=ln(1+2x) is given by
![f(x)=0+2x-2^2\dfrac{x^2}{2!}+2^3\cdot 2! \dfrac{x^3}{3!}+...+(-1)^{(n-1)}(n-1)!\cdot 2^n\dfrac{x^n}{n!}+...\\\\= 0 + 2x -2^2 \dfrac{x^2}{2!}+2^3\dfrac{x^3}{3!}+...+(-1)^{(n-1)}2^{n}\dfrac{x^n}{n}+...\\\\=\sum_{n=1}^{\infty}(-1)^{(n-1)}2^n\dfrac{x^n}{n}](https://tex.z-dn.net/?f=f%28x%29%3D0%2B2x-2%5E2%5Cdfrac%7Bx%5E2%7D%7B2%21%7D%2B2%5E3%5Ccdot%202%21%20%5Cdfrac%7Bx%5E3%7D%7B3%21%7D%2B...%2B%28-1%29%5E%7B%28n-1%29%7D%28n-1%29%21%5Ccdot%202%5En%5Cdfrac%7Bx%5En%7D%7Bn%21%7D%2B...%5C%5C%5C%5C%3D%200%20%2B%202x%20-2%5E2%20%20%5Cdfrac%7Bx%5E2%7D%7B2%21%7D%2B2%5E3%5Cdfrac%7Bx%5E3%7D%7B3%21%7D%2B...%2B%28-1%29%5E%7B%28n-1%29%7D2%5E%7Bn%7D%5Cdfrac%7Bx%5En%7D%7Bn%7D%2B...%5C%5C%5C%5C%3D%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%28-1%29%5E%7B%28n-1%29%7D2%5En%5Cdfrac%7Bx%5En%7D%7Bn%7D)
Answer:
its 45 as3*15 is 45 and 45*1 is 45
plz mark me branliest