# The Exponential of an infinite sum

• WastedGunner
In summary, the conversation discusses a problem in quantum field theory involving combinatorics and connected graphs. The main focus is on proving an identity involving an exponential of an infinite series and the ways to decompose an integer into the sum of integers. The conversation also includes a link to information on partitions in number theory and a discussion on the expansion of the right hand side of the equation. The speaker asks for thoughts on how to prove this identity in a general sense and suggests moving the conversation to a math platform for further discussion.

#### WastedGunner

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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Suggestion: this is a math question. Why not move it there?