How to show whether this double sum is divergent/convergent?

I want to show whether the following series diverge

$\sum_{V=0}^{\infty }\sum_{P=0}^{\infty }\frac{1}{P!V!}\left [ \frac{ig}{6} \int d^{4}x\left ( \frac{1}{i} \frac{\delta }{\delta J(x)}\right )^{3}\right ]^{V}\left [ \frac{i}{2} \int d^{4}yd^{4}zJ(y)J(z)\Delta (y-z)\right ]^{P}$ Where $\frac{\delta }{\delta J(x)}$ is a functional derivative and $\Delta (y-z)$ is a well-behaved complex function.

flag	asked Jan 31 at 21:20 nabil 81●5

Probably this is the wrong forum for the question, not being at a research-level in mathematics... – Gerald Edgar Jan 31 at 22:18

I thought this is not an undergraduate level so I posted it here – nabil Jan 31 at 22:22

The mathematical question is convergence of $$\sum_{V=0}^\infty \sum_{P=0}^\infty \frac{1}{P! V!} a^V b^P$$ which is at an undergraduate level. – Gerald Edgar Jan 31 at 22:32

but what about the functional derivative under V ? It's an operator that acts on $b^{p}$ not a number – nabil Jan 31 at 22:34

Dear nabil, try and follow Gerald Edgar's hints, and your problems will be simpler. Call $a$ and $b$ your expressions into square brackets, whatever they are. What is the sum of the double series? – Pietro Majer Jan 31 at 23:31

show 1 more comment