The answer for this question is 4.
Step-by-step explanation:
The mode is 7 because it is 5 times
In this question (brainly.com/question/12792658) I derived the Taylor series for
about
:
![\mathrm{sinc}\,x=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}](https://tex.z-dn.net/?f=%5Cmathrm%7Bsinc%7D%5C%2Cx%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7B%28-1%29%5Enx%5E%7B2n%7D%7D%7B%282n%2B1%29%21%7D)
Then the Taylor series for
![f(x)=\displaystyle\int_0^x\mathrm{sinc}\,t\,\mathrm dt](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle%5Cint_0%5Ex%5Cmathrm%7Bsinc%7D%5C%2Ct%5C%2C%5Cmathrm%20dt)
is obtained by integrating the series above:
![f(x)=\displaystyle\int\sum_{n=0}^\infty\frac{(-1)^nx^{2n}}{(2n+1)!}\,\mathrm dx=C+\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle%5Cint%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7B%28-1%29%5Enx%5E%7B2n%7D%7D%7B%282n%2B1%29%21%7D%5C%2C%5Cmathrm%20dx%3DC%2B%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7B%28-1%29%5Enx%5E%7B2n%2B1%7D%7D%7B%282n%2B1%29%5E2%282n%29%21%7D)
We have
, so
and so
![f(x)=\displaystyle\sum_{n=0}^\infty\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdisplaystyle%5Csum_%7Bn%3D0%7D%5E%5Cinfty%5Cfrac%7B%28-1%29%5Enx%5E%7B2n%2B1%7D%7D%7B%282n%2B1%29%5E2%282n%29%21%7D)
which converges by the ratio test if the following limit is less than 1:
![\displaystyle\lim_{n\to\infty}\left|\frac{\frac{(-1)^{n+1}x^{2n+3}}{(2n+3)^2(2n+2)!}}{\frac{(-1)^nx^{2n+1}}{(2n+1)^2(2n)!}}\right|=|x^2|\lim_{n\to\infty}\frac{(2n+1)^2(2n)!}{(2n+3)^2(2n+2)!}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft%7C%5Cfrac%7B%5Cfrac%7B%28-1%29%5E%7Bn%2B1%7Dx%5E%7B2n%2B3%7D%7D%7B%282n%2B3%29%5E2%282n%2B2%29%21%7D%7D%7B%5Cfrac%7B%28-1%29%5Enx%5E%7B2n%2B1%7D%7D%7B%282n%2B1%29%5E2%282n%29%21%7D%7D%5Cright%7C%3D%7Cx%5E2%7C%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cfrac%7B%282n%2B1%29%5E2%282n%29%21%7D%7B%282n%2B3%29%5E2%282n%2B2%29%21%7D)
Like in the linked problem, the limit is 0 so the series for
converges everywhere.
the answer to this question is 12 and 60
have a nice day