For the IB Math Sl course I was required to do a "Math exploration". I looked at how the area between the function of e^x and its Taylor Series approximation changes as more terms are added to the series. I attempted to write a function for the area between the two curves in terms of terms added to the series. My end product was:
$$A(k)=e^{10}-1-\sum_{n=1}^{k+1}\frac{1}{n!}10^n$$
However my teachers had a problem with the notation in one of the previous steps, namely:
$$A=\int^{10}_{0} e^{x}-\int^{10}_{0}\sum_{n=0}^{k}\frac{1}{n!}x^n$$
He says that the notation is odd, and that I need to find a way to rewrite this. Any ideas?
Furthermore, when I try to graph this equation the computer cannot process it as there is no way to calculate the value of the sum of a Taylor series for a non-integer number of terms. Is there a way in which I can rewrite the Taylor series to overcome this?
You should take a correct notation for your integration. You have to specify the integration variable. I assume your integration variable is $x$. You should write therefore:
$$A=\int^{10}_{0} e^{x} dx-\int^{10}_{0}\left(\sum_{n=0}^{k}\frac{1}{n!}x^n \right)dx.$$
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The extension of the factorial function to a non-integer argument is the Gamma function. Use the Gamma function and the upper incomplete Gamma function here. See e.g. Wikipedia.
$$\sum_{n=0}^{k}\frac{1}{n!}x^n=\frac{\Gamma(k+1,x)e^{x}}{\Gamma(k+1)}$$
$$\sum_{n=1}^{k+1}\frac{1}{n!}10^n=\frac{\Gamma(k+2,10)e^{10}}{\Gamma(k+2)}-1$$
$$\int\left(\sum_{n=0}^{k}\frac{1}{n!}x^n\right)dx=\sum_{n=0}^{k}\frac{x^{n+1}}{(n+1)!}=\frac{\Gamma(k+2,x)e^x}{\Gamma(k+2)}-1$$