Robert M. Corless
Department of Applied Mathematics
University of Western Ontario
London, Canada

Copyright 2001 by Robert M. Corless
All rights reserved

Useful one-word commands 

Использование монокоманд

Adding terms vs. the finite-difference calculus 

Сложение членов - конечно-разностное счисление

> restart;
> a := add( 1/k, k=1..10 );

a := 7381/2520
> k;

k
> s := sum( 1/k, k=1..10 );

s := 7381/2520
> k;

k
> S := Sum( 1/k, k=1..10 );

S := Sum(1/k,k = 1 .. 10)
> evalf( S );

2.928968254
> add( 1/k, k=1..n );
Error, unable to execute add
> S[n] := sum( 1/k, k=1..n );

S[n] := Psi(n+1)+gamma
> limit( S[n] - log(n), n=infinity );

gamma
> asympt( S[n], n );

ln(n)+gamma+1/2/n-1/12*1/(n^2)+1/120/n^4+O(1/(n^6))...
> restart;
> c := proc( n ) option remember;
-1/(n-1)*add( binomial(n-i+1, i+1)*c(n-i), i=1..n-1 ) end proc;

> c(1) := 1;

c(1) := 1
> seq(c(k),k=1..8);

1, -1, 3/2, -8/3, 31/6, -157/15, 649/30, -9427/210
> Bseries := v -> `if`(v=0,1.,evalf(Sum('c(n)*v^(n-1)', n=1..infinity)));

Bseries := proc (v) options operator, arrow; `if`(v...
> Bseries(0);

1.
> Bseries(0.001);

.9990014973

> Bseries(0.05);

.9534459937

> restart;
> currentdir("C:/books/ess/programs");
> read "cesaro.mpl";
> CesaroSum( (-1)^n, n );

1/2
> normal( CesaroSum( cos(n*x), n ) );

1/2
> part := sum( cos(n*x), n=0..60 );

> plot( part, x=0..2*Pi, y=-1..2, style=POINT, numpoints=1001, symbol=POINT, symbolsize=20, colour=black );

> CesaroSum( sin(n*x), n, 1 );

0
> CesaroSum( x^n/n^2,n, 1 );

> sum( 1/sqrt(n), n=1..k );

sum(1/(sqrt(n)),n = 1 .. k)
> sum( x^n/n^2, n=1..infinity );

polylog(2,x)
> restart;
> alias(alpha=RootOf(z^6+z+1, z));

alpha
> Sum(1/k, k=alpha);

Sum(1/k,k = alpha)
> value( % );

-1
> Product(1/(k+1), k=alpha);

Product(1/(k+1),k = alpha)
> value( % );

1
> product(1/(1+k^2), k=1..n );

1/GAMMA(n+1-I)/GAMMA(n+1+I)*GAMMA(1-I)*GAMMA(1+I)

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