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? – Peter Szilas Jan 29 '20 at 8:43
• Where is the constant? – Fakemistake Jan 29 '20 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 – Siddharth Prakash Jan 29 '20 at 14:28
• @SiddharthPrakash: if you understood my answer, you know that the constant is $\arctan(0)$. – Yves Daoust Jan 29 '20 at 14:29
• I get your answer now thank you very much – Siddharth Prakash Jan 29 '20 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 – Siddharth Prakash Jan 29 '20 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. – Kavi Rama Murthy Jan 29 '20 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. – Isaac Browne Jan 29 '20 at 16:59