# The Exponential of an infinite sum

## Main Question or Discussion Point

I have a problem that arises in quantum field theory. It involves a problem in combinitorics and about the theory of connected graphs.

Essentially, I am trying to prove an identity involving an exponential of an infinite series with the ways to decompose an integer into the sum of integers.

$$\exp\left({\sum_{n=1}^\infty a_n}\right) = \sum_{n=0}^\infty \sum_{partitions} \prod_{i=1}^n \frac{a_i^{n_i}}{n_i ! }$$

Where the partition is over the ways to write n as a sum of integers.

http://en.wikipedia.org/wiki/Partition_(number_theory)

$$\sum_{i=1}^n in_i = n$$

If you expand out the first few terms of the right hand side, it looks good.

n=0 (I'm taking it to be 1 just to avoid confusion)

n=1

$$a_1$$

n=2

$$a_2 + \frac{1}{2!}a_i^2$$

cummulative

$$a_1 + \frac{1}{2!}a_1^2 + a_2$$

n=3

$$a_3 + a_1 a_2 + \frac{1}{3!} a_1^3$$

cummulative

$$a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3+ a_2 + a_1 a_2 + a_3$$

n=4

$$a_4 + a_1 a_3 +\frac{1}{2!} a_2^2 + \frac{1}{2!} a_1^2 a_2 + \frac{1}{4!} a_1^4$$

cummulative

$$a_1 + \frac{1}{2!}a_1^2 + \frac{1}{3!}a_1^3 + \frac{1}{4!}a_1^4 + a_2 + \frac{1}{2!} a_2^2 + a_1 a_2 + \frac{1}{2!}a_1^2 a_2 + a_3 + a_1 a_3 + a_4$$

As you can see, this seems to be systematically giving us the terms of the exponential.

Any thoughts on how to prove this generally?

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