sum

Christopher2222 — Thu, 25 Feb 2010 17:33:45 Z

Your loop can be simplified using sum
So for N=1

sum( (1/k^2)+3 , k=1..30); # or bracket it with evalf for the decimal

For the list you could use seq.

[evalf(seq(sum( (1/k^2)+3 , k = 1..(N*20)), N = 1..30))];

Thanks for the reply

slipper — Thu, 25 Feb 2010 17:53:19 Z

Thanks for the reply Chris,

I don't think i explained mysely to well. I am aware of the sum function in maple, but what i am trying to do is make a loop to simulate this sum function, as in the first post. But how can i get this loop to evaluate all the values of N and output a list? Or can it only be done one at a time? I know its long winded but its just a practice example.

regards

slip

Thanks for the help

slipper — Thu, 25 Feb 2010 20:49:07 Z

Thanks for the help Doug,

Tried a nested loop, but does not seem to evaluate as i was expecting, as in does not give the same values as original equation with N =1, also the print list is quite large, hence the # do not know how to get the (N,sm(N*20) value to show.

restart:sm := 0: n := N*20:

for N from 1 to 30 do

for k from 1 to n do

sm := sm+(1/(k^2+3)):

# print(N,evalf(sm)):

end do:

end do:

regards

small, but important, changes

Doug Meade — Thu, 25 Feb 2010 21:05:20 Z

Close. I see two changes that you need to make.

First, you want each sum to start at 0 - so move the sm:=0 to between the two do statments. Likewise, you want to print the sum only after all n terms have been added - so move the print statement to between the two end do statements. Like this:

restart:
n := N*20:
for N from 1 to 30 do
  sm := 0;
  for k from 1 to n do
    sm := sm+(1/(k^2+3)):
  end do:
  print(N,n,evalf(sm)):
end do:

You do have to be a little careful about the assignment n:=N*20. If this is done when N has a value (as it will have at the end of the do loop) then n will have a specific value. Your code does work correctly assuming the restart is executed immediately before the assignment to n. Personally, I would probably move this assignment inside the first loop (together with the resetting of sm to 0).

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu

Doug, Thanks alot, i can

slipper — Fri, 26 Feb 2010 15:41:18 Z

Doug, Thanks alot, i can see the logic of having sm=0 , and print inside the first loop now, so that each evaluation of N is output to print. Your help greatly appreciated. regards slip

sum and add explained (again)

Doug Meade — Thu, 25 Feb 2010 18:09:16 Z

You need to be a little careful here.

While, in this case, using sum works, using add is preferrable.

Why? What's the difference?

Well, sum is intended for symbolic summation - either a finite (but not explicitly given as a definite number) number of terms or an infinite number of terms. These sums can be evaluated using an algorithm that uses ideas similar to those involved in the Fundamental Theorem of Calculus for definite integrals. That is, these sums are NOT found by the explicit addition of terms.

When you have a finite sum of a definite number of terms the add command just adds the terms.

So, in this case, for N=1, I would use:

add( 1/k^2+3, k=1..20 );
                             667608487627785841
                             ------------------
                             10838475198270720

Moreover, for a general N, the sum command is appropriate. Here's how I might proceed in that case:

S := sum( 1/k^2+3, k=1..20*N );
                                                 1   2
                       60 N - Psi(1, 20 N + 1) + - Pi 
                                                 6    
seq( S, N=1..30 );
         667608487627785841  3471882955795558480635682730386309  
         ------------------, ----------------------------------, 
         10838475198270720    28546916554875489385168794240000   

                              1   2                     1   2  
           180 - Psi(1, 61) + - Pi , 240 - Psi(1, 81) + - Pi , 
                              6                         6      

                               1   2                      1   2  
           300 - Psi(1, 101) + - Pi , 360 - Psi(1, 121) + - Pi , 
                               6                          6      

                               1   2                      1   2  
           420 - Psi(1, 141) + - Pi , 480 - Psi(1, 161) + - Pi , 
                               6                          6      

                               1   2                      1   2  
           540 - Psi(1, 181) + - Pi , 600 - Psi(1, 201) + - Pi , 
                               6                          6      

                               1   2                      1   2  
           660 - Psi(1, 221) + - Pi , 720 - Psi(1, 241) + - Pi , 
                               6                          6      

                               1   2                      1   2  
           780 - Psi(1, 261) + - Pi , 840 - Psi(1, 281) + - Pi , 
                               6                          6      

                               1   2                      1   2  
           900 - Psi(1, 301) + - Pi , 960 - Psi(1, 321) + - Pi , 
                               6                          6      

                                1   2                       1   2  
           1020 - Psi(1, 341) + - Pi , 1080 - Psi(1, 361) + - Pi , 
                                6                           6      

                                1   2                       1   2  
           1140 - Psi(1, 381) + - Pi , 1200 - Psi(1, 401) + - Pi , 
                                6                           6      

                                1   2                       1   2  
           1260 - Psi(1, 421) + - Pi , 1320 - Psi(1, 441) + - Pi , 
                                6                           6      

                                1   2                       1   2  
           1380 - Psi(1, 461) + - Pi , 1440 - Psi(1, 481) + - Pi , 
                                6                           6      

                                1   2                       1   2  
           1500 - Psi(1, 501) + - Pi , 1560 - Psi(1, 521) + - Pi , 
                                6                           6      

                                1   2                       1   2  
           1620 - Psi(1, 541) + - Pi , 1680 - Psi(1, 561) + - Pi , 
                                6                           6      

                                1   2                       1   2
           1740 - Psi(1, 581) + - Pi , 1800 - Psi(1, 601) + - Pi 
                                6                           6    
evalf( % );
61.59616324, 121.6202440, 181.6284056, 241.6325119, 301.6349839, 361.6366354, 

  421.6378167, 481.6387036, 541.6393939, 601.6399466, 661.6403990, 

  721.6407761, 781.6410953, 841.6413690, 901.6416063, 961.6418140, 

  1021.641997, 1081.642160, 1141.642306, 1201.642437, 1261.642556, 

  1321.642664, 1381.642762, 1441.642853, 1501.642936, 1561.643013, 

  1621.643084, 1681.643150, 1741.643211, 1801.643269

The fact that Maple reports the general sum in terms of the Psi function is an artifact of the use of sum's definite summation.

The misuse of sum and the unfamiliarity of add is very common (I know it's been discussed before, and I thought this in a Top Ten Tricks of Maple Experts - but I can't find such a forum or blog entry. Can anyone find this and provide a link to it?).