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Re: Programming an Infinite Sum

• To: mathgroup at smc.vnet.net
• Subject: [mg43321] Re: [mg43278] Programming an Infinite Sum
• From: Dr Bob <drbob at bigfoot.com>
• Date: Sun, 24 Aug 2003 04:55:29 -0400 (EDT)
• References: <200308231209.IAA25182@smc.vnet.net>
• Reply-to: drbob at bigfoot.com
• Sender: owner-wri-mathgroup at wolfram.com

This matches all but the first term of your series --- I think!  (You 
didn't have things in Mathematica notation, so I found it a little 
confusing.)

ClearAll[term, sum]
term[3] = 2/5;
term[n_Integer] /; n > 3 := term[n] = term[n - 1](Prime[n - 1] - 2) 
/Prime[n]
term[n_] := term@Round@n (* for plots *)
sum[3] = term[3];
sum[n_Integer] /; n > 3 := sum[n] = sum[n - 1] + N@term[n]
sum[n_] := sum@Round@n  (* for plots *)
term /@ Range[3, 8]
Short[sum /@ Range[3, 5002], 5]

{2/5, 6/35, 6/77, 54/1001, 54/1547, 810/29393}

{0.4, 0.571429, 0.649351, 0.703297, \[LeftSkeleton]4992\[RightSkeleton],
   0.978568, 0.978569, 0.97857, 0.97857}

(Add 1/5 to get the result with your first term.)

The ratio test is inconclusive:

Plot[(Prime[Round[n]] - 2)/
    Prime[Round[n] + 1], {n, 4, 1000}];

a constant times
1/(n*Log[n]^2.7),

Plot[n*Log[n]^2.7*term[n], {n, 3, 1000}];

but I don't have a ready proof of it, despite the plot.

Likewise, it seems that THAT series converges.

Sum[1/(n*Log[n]^2.7), {n, 2, 100000}]

1.98995

It certainly does if the following is correct:

NSum[1/(n*Log[n]^2), {n, 2, Infinity}]

2.109742801134059 + 0.*I

but I wouldn't bet a gold mine on it, without a theoretical proof.

Bobby

On Sat, 23 Aug 2003 08:09:04 -0400 (EDT), A.S. Haley <ashaley at nccn.net> 
wrote:

> I have a tricky infinite sum over the primes and want to see whether it 
> converges or diverges.  I am having difficulty programming it (in 
> Mathematica 4), and would appreciate some help.  The sum is as follows:
>
> 3/5 + (3/5)(2/7) + (3/5)(5/7)(2/11) + (3/5)(5/7)(9/11)(2/13) + . . .
>
> + [(Prime[3]-2)(Prime[4]-2). . .(Prime[n-1]-2)/(Prime[3])(Prime[4]) . . 
> .(Prime[n-1])] x [2/(Prime[n])]  . . .
>
> and I want to see what happens as n --> Infinity.
>
> Thanks in advance,
>
> Allan Haley
>
>
>



-- 
majort at cox-internet.com
Bobby R. Treat

• Follow-Ups:
  • Re: Re: Programming an Infinite Sum
    • From: Daniel Lichtblau <danl@wolfram.com>

• References:
  • Programming an Infinite Sum
    • From: "A.S. Haley" <ashaley@nccn.net>