yacas comes with a number of rules all defined in the yacas directory of the installed package:
## [1] "/tmp/Rtmpj9GSJ6/Rinst33095a4066/Ryacas/yacas"For example in the sums.rep folder, a number of rules for sums are defined in the code.ys file.
As an example, the fact that \[
\sum_{k = 1}^n (2k-1) = n^2
\] is defined in yacas as
SumFunc(_k,1,_n,2*_k-1, n^2 );and the geometric sum is defined as
SumFunc(_k,0,_n,(r_IsFreeOf(k))^(_k), (1-r^(n+1))/(1-r) );These can be verified:
## [1] "m^2"## [1] "2^(m+1)-1"There are also rules in yacas that are able to let the user change some limits of some sums, e.g. for the geometric sum:
## [1] "2^(m+1)-2"But what about changing the limit of the first sum? I.e. instead of \[
\sum_{k = 1}^n (2k-1) = n^2
\] then know that \[
\sum_{k = 0}^n (2k-1) = -1 + \sum_{k = 1}^n (2k-1) = n^2 - 1 .
\] But what does yacas say?
## [1] "Sum(i,0,m,2*i-1)"We can then add our own rule by:
And then try again:
## [1] "m^2-1"A good source of inspiration for writing custom rules is reading the included rules, but there is a lot to programming in yacas and we refer to yacas’s documentation, specifically the chapter Programming in Yacas.