The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math.symbolic

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: sum(eulerphi(n)/n!,n=1...infinity) = 5*sqrt(Pi)/3 -1 ????
Replies: 6   Last Post: Feb 2, 2013 5:14 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Axel Vogt

Posts: 1,068
Registered: 5/5/07
Re: sum(eulerphi(n)/n!,n=1...infinity) = 5*sqrt(Pi)/3 -1 ????
Posted: Feb 2, 2013 2:56 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 02.02.2013 16:34, wrote:
> Hi,
> How close gets
> sum(eulerphi(n)/n!,n=1...infinity)
> to
> 5*sqrt(Pi)/3 -1
> ????
> Regular (free that is) WolframAlpha gives only few decimal digits via partial sums ("show points" option) and times out without direct evaluation of the sum ...

Note that phi(k) <= k-1 (and equality for prime numers), almost by definition.

For any positive integer we have Sum(phi(k)/k!, k=1 .. infinity)
<= Sum(phi(k)/k!, k=1 .. k0) + Sum(phi(k)/k!, k=k0+1 .. infinity) =
= Sum(phi(k)/k!, k=1 .. k0) + 1/k0!

For the last evaluation one can use Maple or any other system to see that.

Now for k0 = 9 that can be computed quickly as 709099/362880 and that is
strictly smaller than your other term by ~ 1e-5

And for k0 = 20 that error is very small, since 1/k0! --> 0 quite fast.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.