I want to decide whether $$\sum_{n=0}^\infty \int_{[-1,1]^2} \left(1-\frac{x^2+y^2}{2}\right)^{n/2}\lambda_2(d(x,y))$$ is finite or not. $\lambda_2 = 2$-dimensional Lebesgue measure. Attempt:
Since $\left(1-\frac{x^2+y^2}{2}\right)^{n/2}$ is monotone in $n$, by monotone convergence we may exchange sum and integral to get $$\int_{[-1,1]^2}\sum_{n=0}^\infty\left(1-\frac{x^2+y^2}{2}\right)^{n/2} \lambda_2(d(x,y))$$ Since $\left(1-\frac{x^2+y^2}{2}\right)^{n/2} < 1$ almost everywhere we have a geometric series: $$\int_{[-1,1]^2}\frac{1}{1-\sqrt{1-\frac{x^2+y^2}{2}}} \lambda_2(d(x,y))$$ but I don't know if this leads my anywhere. Any hints on how to proceed?
Idea: changing to polar coordinates, the integrand will be $$\frac{r}{1 - \sqrt{1 - r^2/2}},$$ with elementary primitive. Integration limits of $[0,1]\times[0,1]$ in polar coordinates are awkward but feasible...