# Number theory - show divergence of ∑1/p for prime p

1. Jun 13, 2014

### drjohnsonn

1. show that the sum of. The reciprocals of the primes is divergent. Im reposying this here under homework and deleting the inital improperly placed post

2. Theorem i use but dont prove because its assumed thw student has already lim a^1/n = 1.

The gist of the approach I took is that∑1/p = log(e^∑1/p) = log(∏e^1/p) and logx→ ∞ as x→∞.
Proof outline: let ∑1/p = s(x). (...SO I can write this easily on tablet) and note that e^s(x) diverges since e^1/p > 1 for any p and the infinite product where every term exceeds 1 is divergent. Then loge^s(x) diverges as logs as x→∞ would. Thus, since log(e^s(x)= s(x), the sum is found to be divergent

1. The problem statement, all variables and given/known data
Edit: this is wrong and i finished the proof using very little ofwhati tried here so no need to respond

Last edited: Jun 13, 2014
2. Jun 13, 2014

### haruspex

Not so.
Any infinite sum of positive terms ∑an could be written as ln(∏ean)

3. Jun 13, 2014

### drjohnsonn

Indeed. That whopper of an error was pointed out. Can't believe i did that but alas, excitement of an easy solution was blinding.

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