Answer:
18 ft
Step-by-step explanation:
For given power series
the radius of convergence is 2.
For given question,
We have been given a power series ![\sum_{n=0}^{\infty} \frac{(-1)^nx^n}{2^n}](https://tex.z-dn.net/?f=%5Csum_%7Bn%3D0%7D%5E%7B%5Cinfty%7D%20%5Cfrac%7B%28-1%29%5Enx%5En%7D%7B2%5En%7D)
We need to find the radius of convergence of the power series.
We use ratio test to find the radius of convergence of the power series.
Let ![a_n=\frac{(-1)^nx^n}{2^n}](https://tex.z-dn.net/?f=a_n%3D%5Cfrac%7B%28-1%29%5Enx%5En%7D%7B2%5En%7D)
![\Rightarrow a_{n+1}=\frac{(-1)^{n+1}x^{n+1}}{2^{n+1}}](https://tex.z-dn.net/?f=%5CRightarrow%20a_%7Bn%2B1%7D%3D%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7Dx%5E%7Bn%2B1%7D%7D%7B2%5E%7Bn%2B1%7D%7D)
Consider,
![\lim_{n \to \infty}|\frac{a_{n+1}}{a_n} |\\\\= \lim_{n \to \infty} |\frac{\frac{(-1)^{n+1}x^{n+1}}{2^{n+1}}}{ \frac{(-1)^nx^n}{2^n} } |\\\\=\lim_{n \to \infty} |\frac{(-1)^{n+1}x^{n+1}}{2^{n+1}}\times \frac{2^n}{(-1)^nx^n} |\\\\=\lim_{n \to \infty} |\frac{(-1)x}{2} |\\\\=\lim_{n \to \infty}|\frac{-x}{2} |\\\\=\frac{x}{2}](https://tex.z-dn.net/?f=%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%7C%5Cfrac%7Ba_%7Bn%2B1%7D%7D%7Ba_n%7D%20%7C%5C%5C%5C%5C%3D%20%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7B%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7Dx%5E%7Bn%2B1%7D%7D%7B2%5E%7Bn%2B1%7D%7D%7D%7B%20%5Cfrac%7B%28-1%29%5Enx%5En%7D%7B2%5En%7D%20%7D%20%7C%5C%5C%5C%5C%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7Dx%5E%7Bn%2B1%7D%7D%7B2%5E%7Bn%2B1%7D%7D%5Ctimes%20%5Cfrac%7B2%5En%7D%7B%28-1%29%5Enx%5En%7D%20%7C%5C%5C%5C%5C%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%20%7C%5Cfrac%7B%28-1%29x%7D%7B2%7D%20%7C%5C%5C%5C%5C%3D%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%7C%5Cfrac%7B-x%7D%7B2%7D%20%7C%5C%5C%5C%5C%3D%5Cfrac%7Bx%7D%7B2%7D)
By Ratio test, given power series converges at ![|\frac{x}{2} | < 1](https://tex.z-dn.net/?f=%7C%5Cfrac%7Bx%7D%7B2%7D%20%7C%20%3C%201)
⇒ |x| < 2
So, the radius of convergence is 2.
Therefore, for given power series
the radius of convergence is 2.
Learn more about the radius of convergence here:
brainly.com/question/2289050
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Sam's money will double faster.
The formula for compound interest is
A=p(1+r)ˣ, where A is the total amount in the account, p is the principal you invest, r is the interest rate as a decimal, and x is the amount of time.
If we want to represent the money being doubled, we would use 2p for A:
2p=p(1+0.09)ˣ
2p=p(1.09)ˣ
Divide both sides by p:
2p/p = p(1.09)ˣ/p
2 = 1.09ˣ
Using logarithms to solve this,
![\log_{1.09}2=x \\ \\8.04=x](https://tex.z-dn.net/?f=%5Clog_%7B1.09%7D2%3Dx%0A%5C%5C%0A%5C%5C8.04%3Dx)
The formula for simple interest is
I = prt
If we want to find the amount of time to double the principal, this means the amount of interest would be equal to the principal invested:
p = p(0.09)t
Dividing both sides by p,
p/p = p(0.09)t/p
1 = 0.09t
Dividing both sides by 0.09,
1/0.09 = t
11.11 = t
Answer: 8x4+10
Step-by-step explanation: