# The divergence of the series of reciprocals of primes (proof check):

I want to check my attempt at a proof for the divergence of

$$\sum_{n=1}^{\infty} \frac{1}{p_n} \tag{ \star }.$$

We begin with assuming that $$(\star)$$ converges. If $$(\star)$$ converges, there is an integer $$a$$ so that, $$\sum_{j=a+1}^{\infty}\frac{1}{p_j} \lt \frac{1}{b}$$ where $$b>1$$. Note that given any $$b$$ there exists an $$a$$ that satisfies the above inequality. Now, we let $$M = p_1\cdot\cdot\cdot p_a$$ and consider the number $$1+ nM$$ for $$n = 1,2,\dots$$ Any factors of $$1+nM$$ are $$p_i$$ for $$i \geq a+1,a+2,\dots$$ Hence, we can write for each $$g \geq 1$$:

$$\sum_{n=1}^{g} \frac{1}{1+nM} \leq \sum_{x=1}^{\infty}(\sum_{j=a+1}^{\infty}\frac{1}{p_j})^x$$

But on the right hand side, we have a geometric series: $$\sum_{n=1}^{g} \frac{1}{1+nM} \leq \sum_{x=1}^{\infty}(\frac{1}{b})^x$$ Since the geometric series converges, it means that $$\sum_{n=1}^{\infty} \frac{1}{1+nM}$$ converges and is bounded above. But that is a contradiction because if we do the integral test on $$\sum_{n=1}^{\infty} \frac{1}{1+nM}$$, it diverges. Therefore, $$(\star)$$ diverges.

• Alright! So, from the definition of M, we can say: $\frac{1}{1+nM} = \frac{1}{p_{k+1}p_{k+2}\cdots p_{x}}$, so this term must appear somewhere in the expansion of $(\frac{1}{p_{k+1}}+\frac{1}{p_{k+2}}+\frac{1}{p_{k+3}}+\cdots)^x$ - that's how that inequality came to be. – kvmu May 20 '13 at 4:02
• Your statement "Note that as b changes, a changes correspondingly.", though correct, would be better expressed as "given any b, there is an a such that:". This can be valuable in infinite descent proofs, where $a \lt b$,then $a$ satisfies the criterion, so you can apply it again... – Ross Millikan May 20 '13 at 4:36
• Oh, good call - I will edit that right now, but other than that is the proof okay? – kvmu May 20 '13 at 4:43
• Some nitpicks: Clearly you need $b>1$ and you probably meant $i\geq a+1$ – Alex R. May 20 '13 at 5:03
• You might wanna compare your proof to the proof on pg. 18-19 of Tom Apostol's text "Introduction to Analytic Number Theory". I could provide both pages if you like.. – user70962 May 20 '13 at 7:51