Re: Infinite Series Error - Bug?

• To: mathgroup at smc.vnet.net
• Subject: [mg124558] Re: Infinite Series Error - Bug?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 25 Jan 2012 06:59:22 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201201241006.FAA20247@smc.vnet.net>

```Clearly Sum cannot find an explicit formula for the sum of this series.
What seems to occur at some stage looks like underflow to me, but why
Mathematica attempts any numerical computations in this case and if this
is a bug I can't tell. However, Mathematica can tell you two things.

1. That the series is convergent

SumConvergence[q^(-6 + 4*n)/(1 - q^(-5 + 4*n)), n]

q != 0 && Abs[q]^4 < 1

The first condition is curious but it seems right. Secondly, once you
know that:

With[{q = 1/10},
NSum[q^(-6 + 4*n)/(1 - q^(-5 + 4*n)), {n, 0, 10^8}]]

-21.1012

or, if you don't trust machine precision:

With[{q = 1/10},
NSum[q^(-6 + 4*n)/(1 - q^(-5 + 4*n)), {n, 0, Infinity},
WorkingPrecision -> 50]]

-21.101200102001001100190002090100091000992090000000

Andrzej Kozlowski

On 24 Jan 2012, at 11:06, clashton wrote:

> Can anyone explain why evaluating the following expression gives an
> error?
>
> Sum[q^(-6 + 4*n)/(1 - q^(-5 + 4*n)), {n, 0, Infinity}] /. {q -> 0.1}
>
> None of the denominators are 0 (would only happen if -5 + 4*n=0,
which
> is impossible for an integer n, yet I get an error message "Infinite
> expression 1/0 encountered" (several times in fact).
>
> Any help on a work around greatly appreciated.
>
>
> Thanks,
>
> Jimmy
>

```

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