# Does $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x+y)^{1000}}\ dx\,dy$ converge?

I want to check if the following integral diverges or converges.

$$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x+y)^{1000}}\ dx\,dy$$

I'm not sure if it diverges or converges. I attempted to prove divergence, in two ways:

1. Substituting $$x + y = u, x = v$$ we get that the abosolute value of the Jacobian is $$1,$$we get that $$-\infty \le x =v \le \infty$$. However, I'm not sure what are the bounadries on $$u$$. How do I find them? let's say $$a \le u \le b$$. If one of the boundaries of $$u$$ is infinite, then our integral is now $$\int_{-\infty}^{\infty}\int_{a}^{b}e^{-(u)^{1000}}\ dudv,$$ and it diverges. However, again, I'm not sure on the boundaries.

2. Since $$e^{-(x+y)^{1000}} \ge 0$$, we can look at a small enough interval near some $$x_0$$, such that $$x-y$$ is close to $$0$$ - so in that interval, we can say that $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x+y)^{1000}}\ dx\,dy \ge \int_{-\infty}^{\infty}\int_{x_0 - \delta}^{x_0 + \delta}\frac{1}{2}\ dx\,dy,$$ and diverges.

So my questions are: in the first attempt, how to find the boundaries? Are my attempts okay? If not, what's wrong with them?

• In your first approach, the limits on $u$ and $v$ are $\pm \infty$. You need to cover all the points of the plane with the values of $u$ and $v$. Commented Apr 8 at 9:15
• In your second approach, you are actually looking at a small tubular neighborhood around the line $y=-x$, but that’s enough for divergence. Commented Apr 8 at 9:17
• Thanks! Yes, this was my thought - there's always a $y$ for each $x$ that our function is larger than half.
– FNB
Commented Apr 8 at 9:28
• It was in an exam in 202-something. I figured there's nothing special about that number, so I changed it to 1000. I like that number.
– FNB
Commented Apr 8 at 14:36
• Change of variables $u=x+y$ and $v=x-y$ is better IMO, since it represents a rotation as much as $\pi/4$ and a scaling. Commented Apr 8 at 18:57