Sum of geometric series without loop

Hey! For the function geo(r,varargin) i am trying to find the sum of the following

1 + r + r^2 + r^3 + r^4 + ... + r^n

where n = nargin.

I have searched and lurked at every corner both here and in the "help" in Matlab, but i found no way of "writing"/"setting up" the geometric series without loops, which is what i am trying to accomplish. Finding the sum i can do, but i cannot find a way to write above series without too much effort. Any ideas? What obvious built-in function am i missing?

Answer 1

[Geoff Hayes]

I'm not all that clear on why you are using n=nargin instead of just passing n into the function as the second input but maybe there is more to your function than just the series.

You can try using cumprod which returns the cumulative product of each element in the input array. For example,

 cumprod([1 2 3 4 5])

returns

 1     2     6    24   120

where each element (starting from second) is multiplied by the previous element. You can do something similar with r and n as follows

 % create a vector with n elements all identical to r
 v = r*ones(1,n);

 % calculate [r r^2 r^3….r^n]
 v = cumprod(v);

 % sum and add one
 geoSum = 1 + sum(v);

Try the above and see what happens!

[Kenan Hoyt]

It works! Thanks!

[Geoff Hayes]

Great!

Answer 2

[Roger Stafford]

A simpler method:

   result = (1-r^(n+1))/(1-r);

[Kenan Hoyt]

That's really cool. I can't understand how (yet), but it works! Thanks!

[Roger Stafford]

It is easy to see why this works. Call the sum s:

 s = 1 + r + r^2 + r^3 + r^4 + ... + r^n

Then

 s*r = r + r^2 + r^3 + r^4 + r^5 + ... + r^(n+1)

Therefore when you subtract s*r from s, all the terms cancel except the first and last ones:

 s - s*r = 1 - r^(n+1)

Hence if you divide both sides by 1-r you get

 s = (1-r^(n+1))/(1-r)

See the site: http://en.wikipedia.org/wiki/Geometric_series