We can use the fact that, for
,
![\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac1%7B1-x%7D%3D%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20x%5En)
Notice that
![\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1{1-x}\right]=\dfrac1{(1-x)^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Cdfrac1%7B1-x%7D%5Cright%5D%3D%5Cdfrac1%7B%281-x%29%5E2%7D)
so that
![f(x)=\displaystyle\frac5{(1-x)^2}=5\frac{\mathrm d}{\mathrm dx}\left[\sum_{n=0}^\infty x^n\right]](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle%5Cfrac5%7B%281-x%29%5E2%7D%3D5%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20x%5En%5Cright%5D)
![f(x)=\displaystyle5\sum_{n=0}^\infty nx^{n-1}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle5%5Csum_%7Bn%3D0%7D%5E%5Cinfty%20nx%5E%7Bn-1%7D)
![f(x)=\displaystyle5\sum_{n=1}^\infty nx^{n-1}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle5%5Csum_%7Bn%3D1%7D%5E%5Cinfty%20nx%5E%7Bn-1%7D)
![f(x)=\displaystyle5\sum_{n=0}^\infty(n+1)x^n](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle5%5Csum_%7Bn%3D0%7D%5E%5Cinfty%28n%2B1%29x%5En)
By the ratio test, this series converges if
![\displaystyle\lim_{n\to\infty}\left|\frac{(n+2)x^{n+1}}{(n+1)x^n}\right|=|x|\lim_{n\to\infty}\frac{n+2}{n+1}=|x|](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%28n%2B2%29x%5E%7Bn%2B1%7D%7D%7B%28n%2B1%29x%5En%7D%5Cright%7C%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7Bn%2B2%7D%7Bn%2B1%7D%3D%7Cx%7C%3C1)
so the series has radius of convergence
.
The answer is <span>a number that can be written as a ratio of two integers.
I hope this helps!</span>
It’s c because when you round your answer to 3 decimal places is equal to 16
Answer:
< K = 135
Step-by-step explanation:
< K = 180 - < H
= 180 - 45 = 135