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For what positive real values of x does 1/(1^x) + 1/(2^x) + 1/(3^x)... converge? watch

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    I think that's called the Zeta function not sure though, where it goes 1/(1^x) + 1/(2^x) + 1/(3^x) 1+(4^x)... all the way to infinity.
    I know it definitely converges for x>2 and it diverges for x=1, but what about the numbers between 1 and 2? I'm not that sure here. I'm assuming low values like x^1.00000001 diverge as well, but it's just a guess, since the reciprocals of primes diverge.
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    (Original post by CancerousProblem)
    I think that's called the Zeta function not sure though, where it goes 1/(1^x) + 1/(2^x) + 1/(3^x) 1+(4^x)... all the way to infinity.
    I know it definitely converges for x>2 and it diverges for x=1, but what about the numbers between 1 and 2? I'm not that sure here. I'm assuming low values like x^1.00000001 diverge as well, but it's just a guess, since the reciprocals of primes diverge.
    When x=2 it converges to pi squared over 6. Basel problem.


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    (Original post by physicsmaths)
    When x=2 it converges to pi squared over 6. Basel problem.


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    I've done my homework and I know that, and how Euler got the result as well. Doesn't answer my question though.
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    (Original post by CancerousProblem)
    I think that's called the Zeta function not sure though, where it goes 1/(1^x) + 1/(2^x) + 1/(3^x) 1+(4^x)... all the way to infinity.
    I know it definitely converges for x>2 and it diverges for x=1, but what about the numbers between 1 and 2? I'm not that sure here. I'm assuming low values like x^1.00000001 diverge as well, but it's just a guess, since the reciprocals of primes diverge.
    The sum you gave (absolutely) convergences whenever x>1; x\in \mathbb{R}

    The zeta function \zeta(z)=\sum_{n=1}^{\infty} \frac{1}{n^z} converges (absolutely) whenever the real part of z ,Re(z)>1; z \in \mathbb{C}. However, you can write it in another way to give convergence at more values.
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    (Original post by CancerousProblem)
    I've done my homework and I know that, and how Euler got the result as well. Doesn't answer my question though.
    Alright, chill out.


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    (Original post by CancerousProblem)
    I've done my homework and I know that, and how Euler got the result as well. Doesn't answer my question though.
    You said specifically x>2 not x>=2.


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    (Original post by CancerousProblem)
    I've done my homework and I know that, and how Euler got the result as well. Doesn't answer my question though.
    "Doing your homework" should include googling the phrase "convergence of zeta function"!


    You'll find the answer to your question and at least one link to a proof within the first page of relevant links
 
 
 
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