Help with capital sigma notation please.

[rustynail]

I was playing a bit with the Riemann Zeta function, and have been struggling with some notation problems.

The function is defined as follows

\zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

where s \in \mathbb{C}

we know that

n^s = exp(s\;ln\;n)

so I can write

\zeta (s) = \sum_{n=1}^{\infty} \frac{1}{exp(s\;ln\;n)}

but since

\frac{1}{exp(s\;ln\;n)} = 1 + \frac{1!}{(s\;ln\;n)} + \frac{2!}{(s\;ln\;n)^2} + ... = <br /> 1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n}

how can I write this ''sum within a sum''? ζ(s) here, if I am correct, would be an infinite sum of terms which are infinite sums.

Thank you for taking the time to help!edit :

Could I say

1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n} = a_k

then

\zeta (s) = \sum_{k=1}^{\infty} a_k

Does that make any sense?

 

[D H]

but since

\frac{1}{exp(s\;ln\;n)} = 1 + \frac{1!}{(s\;ln\;n)} + \frac{2!}{(s\;ln\;n)^2} + ... = <br /> 1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n}

No.

What you can say is that
\exp (s \ln n) = \sum_{r=0}^{\infty} \frac{(s\ln n)^r}{r!}
What you did, simply inverting each term on the right hand side to get 1/\exp(s\ln n), is invalid.

This is what you need to use for 1/\exp(s\ln n):
\frac 1{\exp(s\ln n)} = \exp (-s \ln n) =<br /> \sum_{r=0}^{\infty} \frac{(-s\ln n)^r}{r!} =<br /> \sum_{r=0}^{\infty} \frac{(-1)^r(s\ln n)^r}{r!}

 

[DonAntonio]

I was playing a bit with the Riemann Zeta function, and have been struggling with some notation problems.

The function is defined as follows

\zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

where s \in \mathbb{C}

This only makes sense for \,Re(s)&gt;1\,...careful!

DonAntonio

we know that

n^s = exp(s\;ln\;n)

so I can write

\zeta (s) = \sum_{n=1}^{\infty} \frac{1}{exp(s\;ln\;n)}

but since

\frac{1}{exp(s\;ln\;n)} = 1 + \frac{1!}{(s\;ln\;n)} + \frac{2!}{(s\;ln\;n)^2} + ... = <br /> 1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n}

how can I write this ''sum within a sum''? ζ(s) here, if I am correct, would be an infinite sum of terms which are infinite sums.

Thank you for taking the time to help!


edit :

Could I say

1 + \sum_{n=1}^{\infty} \frac{n!}{(s\;ln\;n)^n} = a_k

then

\zeta (s) = \sum_{k=1}^{\infty} a_k

Does that make any sense?

 

[rustynail]

Thank you DH for the help, and DonAntonio for your rigor, I need to work on that.

But in this part,

\sum_{r=0}^{\infty} \frac{(-1)^r(s\ln n)^r}{r!}

Don't we need to to take the sum of all terms with both n and r, from 1 to ∞? Could I write it as

\sum_{n=1}^{\infty} \; \sum_{r=0}^{\infty} \frac{(-1)^r(s\ln n)^r}{r!} ??