Yes, they are equivalent
6+7-5=8. -5+10+3=8
The answer is actually D. The limit on the left side is computing the derivative at x = a. The right side value of 7 tells us that f ' (a) = 7
Answer:
1.28
Step-by-step explanation:
Assuming in 8a2 the '2' represents an exponent, you answer should be
1.28
a. Recall that
![\displaystyle\int\frac{\mathrm dx}{1-x}=-\ln|1-x|+C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20dx%7D%7B1-x%7D%3D-%5Cln%7C1-x%7C%2BC)
For
, we have
![\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)
By integrating both sides, we get
![\displaystyle-\ln(1-x)=C+\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}](https://tex.z-dn.net/?f=%5Cdisplaystyle-%5Cln%281-x%29%3DC%2B%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D)
If
, then
![\displaystyle-\ln1=C+\sum_{n=0}^\infty\frac{0^{n+1}}{n+1}\implies 0=C+0\implies C=0](https://tex.z-dn.net/?f=%5Cdisplaystyle-%5Cln1%3DC%2B%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7B0%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D%5Cimplies%200%3DC%2B0%5Cimplies%20C%3D0)
so that
![\displaystyle\ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cln%281-x%29%3D-%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7Bx%5E%7Bn%2B1%7D%7D%7Bn%2B1%7D)
We can shift the index to simplify the sum slightly.
![\displaystyle\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cln%281-x%29%3D-%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7Bx%5En%7Dn)
b. The power series for
can be obtained simply by multiplying both sides of the series above by
.
![\displaystyle x\ln(1-x)=-\sum_{n=1}^\infty\frac{x^{n+1}}n](https://tex.z-dn.net/?f=%5Cdisplaystyle%20x%5Cln%281-x%29%3D-%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac%7Bx%5E%7Bn%2B1%7D%7Dn)
c. We have
![\ln2=-\dfrac\ln12=-\ln\left(1-\dfrac12\right)](https://tex.z-dn.net/?f=%5Cln2%3D-%5Cdfrac%5Cln12%3D-%5Cln%5Cleft%281-%5Cdfrac12%5Cright%29)
![\displaystyle\implies\ln2=\sum_{n=1}^\infty\frac1{n2^n}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cimplies%5Cln2%3D%5Csum_%7Bn%3D1%7D%5E%5Cinfty%5Cfrac1%7Bn2%5En%7D)
100 centimeters is equal a meter so you can say 345 centimetres is the same as 3,45 metres
i hope i helped you somehow