Assuming the series is
![\displaystyle\sum_{n\ge1}\frac{x^n}{2n-1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Csum_%7Bn%5Cge1%7D%5Cfrac%7Bx%5En%7D%7B2n-1%7D)
The series will converge if
![\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{2(n+1)-1}}{\frac{x^n}{2n-1}}\right|](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7B2%28n%2B1%29-1%7D%7D%7B%5Cfrac%7Bx%5En%7D%7B2n-1%7D%7D%5Cright%7C%3C1)
We have
![\displaystyle\lim_{n\to\infty}\left|\frac{\frac{x^{n+1}}{2(n+1)-1}}{\frac{x^n}{2n-1}}\right|=|x|\lim_{n\to\infty}\frac{\frac1{2n+1}}{\frac1{2n-1}}=|x|-\lim_{n\to\infty}\frac{2n-1}{2n+1}=|x|](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7B2%28n%2B1%29-1%7D%7D%7B%5Cfrac%7Bx%5En%7D%7B2n-1%7D%7D%5Cright%7C%3D%7Cx%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%5Cfrac1%7B2n%2B1%7D%7D%7B%5Cfrac1%7B2n-1%7D%7D%3D%7Cx%7C-%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B2n-1%7D%7B2n%2B1%7D%3D%7Cx%7C%3C1)
So the series will certainly converge if
![-1](https://tex.z-dn.net/?f=-1%3Cx%3C1)
, but we also need to check the endpoints of the interval.
If
![x=1](https://tex.z-dn.net/?f=x%3D1)
, then the series is a scaled harmonic series, which we know diverges.
On the other hand, if
![x=-1](https://tex.z-dn.net/?f=x%3D-1)
, by the alternating series test we can show that the series converges, since
![\left|\dfrac{(-1)^n}{2n-1}\right|=\dfrac1{2n-1}\to0](https://tex.z-dn.net/?f=%5Cleft%7C%5Cdfrac%7B%28-1%29%5En%7D%7B2n-1%7D%5Cright%7C%3D%5Cdfrac1%7B2n-1%7D%5Cto0)
and is strictly decreasing.
So, the interval of convergence for the series is
![-1\le x](https://tex.z-dn.net/?f=-1%5Cle%20x%3C1)
.
Answer:
x=9.5
Step-by-step explanation:
In order to find the answer, we know that 6x + 4x - 5 = 90
Next we want to combine like terms: 10x - 5 = 90.
Now we have to add 5 to 90 in order to get x on its own. 10x = 95
Now solve for x by dividing 95 by 10: x=9.5
3) 2,000.376 inches of water
4) 30.375 for 1 box, 91.125 inches for all three
5) 4,536 feet
6) 24 feet
Answer:
B. disability insurance
They also provide life and health insurance.
BTW this is not a math question.
Step-by-step explanation:
Answer: x=4
Step-by-step explanation: write down 11x-2+9x+1=19x+3, then combine the like terms... 20x-1=19x+3, then subtract 19x from both sides which will give you x-1=3, and 4-1=3 so x=4