# Integration of Series Expansion of Exponential Function

There is a theorem that the power series can be integrated term by term over any closed and bounded interval contained in the interval of convergence.

I assume I am using this theorem when I obtain the series expansion of $$\tan^{-1} (x)$$ by integrating each term in the power series expansion of $$\frac{1}{1+x^2}$$. But this theorem fails me when I apply it to the power series expansion of $$\ e^x$$

Since $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}.....(1)$$ Now integrating both sides $$\int e^xdx=\int 1\ dx+\int x\ dx+\int \frac{x^2}{2!}\ dx....$$ $$e^x=x+\frac{x^2}{2!}+\frac{x^3}{3!}....(2)$$ I belive eq (1) and eq(2) are different. I know that I haven't added the constant of integration. That is because as I mentioned earlier the series expansion of $$tan^{-1}x$$ i obtained by integration of series expansion of $$\frac{1}{1+x^2}$$ and no integration constant was added there. And I know that expansion is correct because I got that from "tom apostol calculus 1" $$\int\frac{1}{1+x^2}=\int1-\int x^2+\int x^4-\int x^6.... (3)$$ $$tan^{-1}x=x-\frac{x^3}{3}+\frac{x^5}{5}......(4)$$ So how is eq (4) supposed to be right if eq(2) is wrong.

• Integration constant? Jan 29, 2020 at 8:43
• Where is the constant? Jan 29, 2020 at 8:44

$$\int_0^x e^tdt=e^x\color{red}{-1}$$ matches your term-wise integration. (While $$\displaystyle\int_0^x t^ndt=\dfrac{x^{n+1}}{(n+1)!}$$, constant term is $$0$$.)

• the series expansion of tan inverse is obtained by a similar integration of an expansion which does not involve adding a constant. I have explained it in detail in an edit. Could you please read it and explain why the first one is wrong but the second one right Jan 29, 2020 at 14:28
• @SiddharthPrakash: if you understood my answer, you know that the constant is $\arctan(0)$.
– user65203
Jan 29, 2020 at 14:29
• I get your answer now thank you very much Jan 29, 2020 at 14:38

You have to add a constant when you take anti-derivative. The two expression you got for $$e^{x}$$ are some except for the constant term so there is no contradiction.

• the series expansion of tan inverse is obtained by a similar integration of an expansion which does not involve adding a constant. I have explained it in detail in an edit. Could you please read it and explain why the first one is wrong but the second one right Jan 29, 2020 at 14:27
• @SiddharthPrakash In the second case you got a correct result by a wrong method! The constant of integration can happen to be $0$ in some cases but it is wrong to omit the constant. Jan 29, 2020 at 23:23

You're forgetting the +C!

• As for the inverse tangent expansion you found, it does involve adding a constant, it just happens that $C=0$. You have just found the right answer by accident in this case. Jan 29, 2020 at 16:59