Aren't there infinitely many primes?

1. Aug 2, 2015

PcumP_Ravenclaw

1. The problem statement, all variables and given/known data
sn= 1/n if n is a prime number; sn = 0 if n is not prime.

2. Relevant equations

3. The attempt at a solution
We know that there are infinitely many prime numbers as n tends to infinity so why is $\frac{1}{(prime number)}$ not equal to zero when n is a prime number?

2. Aug 2, 2015

SammyS

Staff Emeritus
It looks like Sn is a term in a series with a particular definition. That definition is what you have given.

Is this a fragment from a problem you have been given.

3. Aug 3, 2015

PcumP_Ravenclaw

The question to this problem is as below.

For each of the sequences defined in below, state whether or not it tends
to a limit. If a sequence has a limit, make an $(\epsilon,N )$ table, taking
$\epsilon = 0.001$ and any other values that you like.

4. Aug 3, 2015

Ray Vickson

When n is an integer (prime or not), 1/n is not zero; however, it is small if n is large. In fact, 1/n is never 0, but $0 = \lim_{n \to \infty} 1/n$.

5. Aug 3, 2015

HallsofIvy

Staff Emeritus
Because there are an infinite number of primes, the limit as n goes to infinity of your sequence is 0. Now, $\frac{1}{primenumber}$ is a different sequence all though its limit is also 0.

Your series "$a_n$" is 0, 1/2, 1/3, 0, 1/5, 0, 1/7, 0, 0, 0, 1/11, ...
Your series "$\frac{1}{primenumber}$" is 1/2, 1/3 ,1/5, 1/7, 1/11, 1/13, 1/17, ...

6. Aug 4, 2015

PcumP_Ravenclaw

how do you know that for the an sequence non-prime terms for n equal 0? its not given in the question.

7. Aug 4, 2015

SammyS

Staff Emeritus
It is given in the definition of the sequence.

It seems you don't understand the word "defined".

The definition of this sequence is:

sn = 1/n, if n is prime
sn = 0, if n is not prime​

1 is not prime, so s1 = 0
2 is prime, so s2 = 1/2
3 is prime, so s3 = 1/3
4 is not prime, so s4 = 0
5 is prime, so s5 = 1/5
6 is not prime, so s6 = 0
7 is prime, so s7 = 1/7
8 is not prime, so s8 = 0
9 is not prime, so s9 = 0
10 is not prime, so s10 = 0
11 is prime, so s11 = 1/11
...

Get it ?

8. Aug 4, 2015

PcumP_Ravenclaw

yes right. Its given in the question. danke!! Haste leads to waste!!