What Went Wrong in My Integration by Parts?

[Username007]

Here I used integration by parts to try to solve an integral (I got it wrong, it seems), I know this has no "simple" solution, but, can anyone explain me exactly what did I do wrong? Here is what I did:


\int\frac{e^x}{x}dx=\frac{e^x}{x}-(-1)\int\frac{e^x}{x^2}dx=\frac{e^x}{x}+1(\frac{e^x}{x^2}-(-2)\int\frac{e^x}{x^3}dx)=\frac{e^x}{x}+1!(\frac{e^x}{x^2}+2!\frac{e^x}{x^3}+3!\int\frac{e^x}{x^4}dx)=...=0!\frac{e^x}{x}+1!\frac{e^x}{x^2}+2!\frac{e^x}{x^3}+3!\frac{e^x}{x^4}+...

I just used integration by parts on the "remaining" integrals so I could produce a series, but the series is just...wrong. I would appreciate if you told me what is the error in my reasoning.
thanks.

 

[tiny-tim]

Welcome to PF!

Hi Username007! Welcome to PF!

the integration by parts looks ok to me …

we can check by using the https://www.physicsforums.com/library.php?do=view_item&itemid=353" to differentiate e^x(∑_n=0n!/xⁿ⁺¹),

we get e^x(∑_n=0n!/xⁿ⁺¹ - ∑_n=1n!/xⁿ⁺¹) = e^x/x

(but does it converge? )

 

[Username007]

It doesn't, that's the point. Why can I use integration by parts and "say" that \int\frac{e^x}{x}dx equals a sum that does not converge?

 

[Username007]

And, thanks :)

 

[tiny-tim]

Hi Username007!

It doesn't, that's the point. Why can I use integration by parts and "say" that \int\frac{e^x}{x}dx equals a sum that does not converge?

integration by parts works fine so long as the ∑ is finite …

∫ e^x/x dx

= e^x(∑_n=0^k-1 n!/xⁿ⁺¹ - n! ∫ e^x/xⁿ⁺¹ dx​

unfortunately, although we can usually rely on the final integral converging to zero as k -> ∞, in this case it doesn't!

 

[Username007]

So, if the result of integration does not converge we can't establish the equality? (Thanks for you patience btw ^_^)

 

[tiny-tim]

yes