Sequences and Series

Command/Example Description seq(f(n), n=i..j); Examples: seq(1/n,n=1..10); a:=seq(k!,k=3..10); a[3]; a[8]; seq(b[m]=sin(m*Pi/2),m=0..8); pt:=seq([n,f(n)],n=1..8); pt[2]; It creates a finite sequence of values f(i), f(i+1), · · ·, and f(j), where f(n) is a maple function and i ≤ j are integers. A sequence of points on the graph of y = f(x) can be obtained using: > seq([n,f(n)], n=i..j); sum(f(n), n=i..j) Examples: sum(n^2, n=-1..10); sum((1/2)^n,n=0..infinity); sum(c/k^2,k=1..infinity); S:=n->sum(k,k=1..n); S(n); S(8); f:=x->sum(x^n/n!,n=0..infinity); f(x); f(1); f(-1); It creates and evaluates a finite or infinite sum, that is, series ∑j n=i f(n), where f(n) is a maple function or expression and i ≤ j can be integers, variables, or infinity. For a finite or convergent infinite series, it automatically evaluates the sum and returns a value or formula. If you don’t want the automatic evaluation, use Sum instead of sum. for n from i to j do...end do; Examples: for n from 5 to 10 do c[n]:=1/n end do; for n from 0 to 9 do d[2*n]:=1; d[2*n+1]:=0 end do; s[1]:=1; for n from 2 to 8 do s[n]:= s[n-1]+n end do; A typical for-loop (for and do statement) used in general programming languages. It executes whatever between ‘‘do’’ and ‘‘end do’’ repeatedly for a counted number of times (‘‘for n from i to j’’). It hence can be used to work with sequences in much more general ways than what the command seq could. • A link to the SequenceDrill maplet can be found on the course website: