closed form for a sum

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3 total messages Started by Bart E. Goddard Tue, 12 Jan 1993 18:08

#2672

Author: Bart E. Goddard
Date: Tue, 12 Jan 1993 18:08

12 lines
877 bytes

Evaluate: 
        \infinity 
          ___
         \        sin(n)/n
         /___
         n = 1
The solution is nice.
--
Bart Goddard                            |Behold, thy King cometh unto thee,
Math. Dept. Rose-Hulman Inst. of Tech.  |meek, and sitting upon an ass, and
goddard@nextwork.rose-hulman.edu        |a colt, the foal of an ass.

Re: closed form for a sum (SPOILER)

#2680

Author: Daniel A. Asimov
Date: Wed, 13 Jan 1993 02:31

30 lines
1449 bytes

In article <1iv1enINNnq6@master.cs.rose-hulman.edu> goddard@NeXTwork.Rose-Hulman.Edu (Bart E. Goddard) writes:
>
>Evaluate: 
>        \infinity 
>          ___
>         \        sin(n)/n
>         /___
>         n = 1
>The solution is nice.
>--
>Bart Goddard                            |Behold, thy King cometh unto thee,
>Math. Dept. Rose-Hulman Inst. of Tech.  |meek, and sitting upon an ass, and
>goddard@nextwork.rose-hulman.edu        |a colt, the foal of an ass.

The sum is (pi - 1)/2.

(Sketch):  Identify the sum as the imaginary part of sum (exp(in)/n).
Consider sum (z^n/n) and e.g. by differentiating this, identify it with
log(1/(1-z)). Now plug in exp(in) for z, take the imaginary part, and
use the half-angle formula to see that it is (pi-1)/2.

Dan Asimov
Mail Stop T045-1
NASA Ames Research Center
Moffett Field, CA 94035-1000

asimov@nas.nasa.gov
(415) 604-4799

Re: closed form for a sum

#2682

Author: Frank Pinto
Date: Wed, 13 Jan 1993 04:38

9 lines
607 bytes

If you mean by evaluation "the limit"
then "it is " 0

Since sin can onlybe from -1, 1
the denominator will grow much "faster"
which bring the ratio 
to a very small number, namely = 0

Frank Pinto

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