Answer:
$8
Step-by-step explanation:
20% is 1/5, so you can basically divide $40 by 5, which is $8.
4x-(5-x)=8[4-(8+2x-1)]
answer: x= -19/21
Answer:
1) 2/3, two triangle segments
2) 1/7, one triangle segment
3) 3/9, three triangle segments left
really, you could do this by subtracting the top number, the numerator, because these visual diagrams have the same bottom number, or denominator, but for now, just subtract the segments. hope this helps, good luck.
I suppose you mean
![f(x)=\cos(x^2)](https://tex.z-dn.net/?f=f%28x%29%3D%5Ccos%28x%5E2%29)
Recall that
![\cos x=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{x^{2n}}{(2n)!}](https://tex.z-dn.net/?f=%5Ccos%20x%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bx%5E%7B2n%7D%7D%7B%282n%29%21%7D)
which converges everywhere. Then by substitution,
![\cos(x^2)=\displaystyle\sum_{n=0}^\infty(-1)^n\frac{(x^2)^{2n}}{(2n)!}=\sum_{n=0}^\infty(-1)^n\frac{x^{4n}}{(2n)!}](https://tex.z-dn.net/?f=%5Ccos%28x%5E2%29%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7B%28x%5E2%29%5E%7B2n%7D%7D%7B%282n%29%21%7D%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bx%5E%7B4n%7D%7D%7B%282n%29%21%7D)
which also converges everywhere (and we can confirm this via the ratio test, for instance).
a. Differentiating the Taylor series gives
![f'(x)=\displaystyle4\sum_{n=1}^\infty(-1)^n\frac{nx^{4n-1}}{(2n)!}](https://tex.z-dn.net/?f=f%27%28x%29%3D%5Cdisplaystyle4%5Csum_%7Bn%3D1%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bnx%5E%7B4n-1%7D%7D%7B%282n%29%21%7D)
(starting at
because the summand is 0 when
)
b. Naturally, the differentiated series represents
![f'(x)=-2x\sin(x^2)](https://tex.z-dn.net/?f=f%27%28x%29%3D-2x%5Csin%28x%5E2%29)
To see this, recalling the series for
, we know
![\sin(x^2)=\displaystyle\sum_{n=0}^\infty(-1)^{n-1}\frac{x^{4n+2}}{(2n+1)!}](https://tex.z-dn.net/?f=%5Csin%28x%5E2%29%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5E%7Bn-1%7D%5Cfrac%7Bx%5E%7B4n%2B2%7D%7D%7B%282n%2B1%29%21%7D)
Multiplying by
gives
![-x\sin(x^2)=\displaystyle2x\sum_{n=0}^\infty(-1)^n\frac{x^{4n}}{(2n+1)!}](https://tex.z-dn.net/?f=-x%5Csin%28x%5E2%29%3D%5Cdisplaystyle2x%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bx%5E%7B4n%7D%7D%7B%282n%2B1%29%21%7D)
and from here,
![-2x\sin(x^2)=\displaystyle 2x\sum_{n=0}^\infty(-1)^n\frac{2nx^{4n}}{(2n)(2n+1)!}](https://tex.z-dn.net/?f=-2x%5Csin%28x%5E2%29%3D%5Cdisplaystyle%202x%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7B2nx%5E%7B4n%7D%7D%7B%282n%29%282n%2B1%29%21%7D)
![-2x\sin(x^2)=\displaystyle 4x\sum_{n=0}^\infty(-1)^n\frac{nx^{4n}}{(2n)!}=f'(x)](https://tex.z-dn.net/?f=-2x%5Csin%28x%5E2%29%3D%5Cdisplaystyle%204x%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28-1%29%5En%5Cfrac%7Bnx%5E%7B4n%7D%7D%7B%282n%29%21%7D%3Df%27%28x%29)
c. This series also converges everywhere. By the ratio test, the series converges if
![\displaystyle\lim_{n\to\infty}\left|\frac{(-1)^{n+1}\frac{(n+1)x^{4(n+1)}}{(2(n+1))!}}{(-1)^n\frac{nx^{4n}}{(2n)!}}\right|=|x|\lim_{n\to\infty}\frac{\frac{n+1}{(2n+2)!}}{\frac n{(2n)!}}=|x|\lim_{n\to\infty}\frac{n+1}{n(2n+2)(2n+1)}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7D%5Cfrac%7B%28n%2B1%29x%5E%7B4%28n%2B1%29%7D%7D%7B%282%28n%2B1%29%29%21%7D%7D%7B%28-1%29%5En%5Cfrac%7Bnx%5E%7B4n%7D%7D%7B%282n%29%21%7D%7D%5Cright%7C%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%5Cfrac%7Bn%2B1%7D%7B%282n%2B2%29%21%7D%7D%7B%5Cfrac%20n%7B%282n%29%21%7D%7D%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7Bn%2B1%7D%7Bn%282n%2B2%29%282n%2B1%29%7D%3C1)
The limit is 0, so any choice of
satisfies the convergence condition.
Answer:
the answer is ![x=3](https://tex.z-dn.net/?f=x%3D3)
Step-by-step explanation:
Try using Symbolab, I use it all the time it gives the correct answer and it gives good explanations.
hope this helps :)