Kathy earned $300 mowing lawns last summer. She spent all her earnings on a new CD player and new CDs. If Kathy paid $90 for the
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You can use the root test here. The series will converge if
![L=\displaystyle\lim_{n\to\infty}\sqrt[n]{\frac{(4-x)^n}{4^n+9^n}}](https://tex.z-dn.net/?f=L%3D%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Csqrt%5Bn%5D%7B%5Cfrac%7B%284-x%29%5En%7D%7B4%5En%2B9%5En%7D%7D%3C1)
You have
![L=\displaystyle\lim_{n\to\infty}\sqrt[n]{\frac{(4-x)^n}{4^n+9^n}}=|4-x|\lim_{n\to\infty}\frac1{\sqrt[n]{4^n+9^n}}](https://tex.z-dn.net/?f=L%3D%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Csqrt%5Bn%5D%7B%5Cfrac%7B%284-x%29%5En%7D%7B4%5En%2B9%5En%7D%7D%3D%7C4-x%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac1%7B%5Csqrt%5Bn%5D%7B4%5En%2B9%5En%7D%7D)
Notice that
![\dfrac1{\sqrt[n]{4^n+9^n}}=\dfrac1{\sqrt[n]{9^n}\sqrt[n]{1+\left(\frac49\right)^n}}=\dfrac1{9\sqrt[n]{1+\left(\frac49\right)^n}}](https://tex.z-dn.net/?f=%5Cdfrac1%7B%5Csqrt%5Bn%5D%7B4%5En%2B9%5En%7D%7D%3D%5Cdfrac1%7B%5Csqrt%5Bn%5D%7B9%5En%7D%5Csqrt%5Bn%5D%7B1%2B%5Cleft%28%5Cfrac49%5Cright%29%5En%7D%7D%3D%5Cdfrac1%7B9%5Csqrt%5Bn%5D%7B1%2B%5Cleft%28%5Cfrac49%5Cright%29%5En%7D%7D)
so as
, you have 
, which means you end up with
This is the interval of convergence. The radius of convergence can be determined by finding the half-length of the interval, or by solving the inequality in terms of
so that 
is the ROC. You get
You have
Notice that
so as
This is the interval of convergence. The radius of convergence can be determined by finding the half-length of the interval, or by solving the inequality in terms of