       Re: A curious result?

• To: mathgroup at smc.vnet.net
• Subject: [mg91507] Re: A curious result?
• From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
• Date: Sun, 24 Aug 2008 07:07:16 -0400 (EDT)
• Organization: The Open University, Milton Keynes, UK
• References: <g8o89u\$l8q\$1@smc.vnet.net>

```amzoti wrote:

> a. Sum[1/n^1.001, {n, 1 , Infinity}] = 1000.58
>
> b. Integrate[1/n^(1.001), {n, 1, Infinity}] = 1000

Only curious when one assumes that Sum[] and Integrate[] use the same
algorithm.

In:= Sum[1/n^1.001, {n, 1, Infinity}]

Out= 1000.58

In:= Sum[1/n^(1001/1000), {n, 1, Infinity}]
% // N

Out= Zeta[1001/1000]

Out= 1000.58

In:= Zeta[1.001]

Out= 1000.58

In:= Integrate[1/n^(1.001), {n, 1, Infinity}]

Out= 1000.

In:= Integrate[1/n^(1001/1000), {n, 1, Infinity}]

Out= 1000

In:= NIntegrate[1/n^(1.001), {n, 1, Infinity}]

During evaluation of In:= NIntegrate::slwcon: Numerical \
integration converging too slowly; suspect one of the following: \
singularity, value of the integration is 0, highly oscillatory \
integrand, or WorkingPrecision too small. >>

During evaluation of In:= NIntegrate::ncvb: NIntegrate failed to \
converge to prescribed accuracy after 9 recursive bisections in n \
near {n} = {8.16907*10^224}. NIntegrate obtained 1000.6295219735948` \
and 5.458904209437807` for the integral and error estimates. >>

Out= 1000.63

Regards,
-- Jean-Marc

```

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