ex11-12-13.mw

> Exercice 11

> Sum(1/(k^2+(n-k)^2),k=0..n)=series(Sum(1/(k^2+(n-k)^2),k=0..n),n=infinity,2);

Sum(1/(k^2+(n-k)^2), k = 0 .. n) = 1/2*Pi/n+O(1/n^2)

>

> Exercice 12

> Limit(Sum(n/(n^2+k^2),k=1..n),n=infinity)=limit(sum(n/(n^2+k^2),k=1..n),n=infinity);

Limit(Sum(n/(n^2+k^2), k = 1 .. n), n = infinity) = 1/4*Pi

> Limit(Sum(1/sqrt(n^2+k^2),k=1..n),n=infinity)=limit(sum(1/sqrt(n^2+k^2),k=1..n),n=infinity);

Limit(Sum(1/(n^2+k^2)^(1/2), k = 1 .. n), n = infinity) = ln(1+2^(1/2))

> Limit((1/n)*Sum(ln(k+n)-ln(n),k=1..n),n=infinity)=limit((1/n)*sum(ln(k+n)-ln(n),k=1..n),n=infinity);

Limit((Sum(ln(k+n)-ln(n), k = 1 .. n))/n, n = infinity) = 2*ln(2)-1

>

>

> Exercice 13

> Limit(((2*n)!/(n!*n^n))^(1/n),n=infinity)=limit(((2*n)!/(n!*n^n))^(1/n),n=infinity);

Limit((factorial(2*n)/(factorial(n)*n^n))^(1/n), n = infinity) = 4*exp(-1)

> Limit((1/n)*(Product(n+k,k=1..n))^(1/n),n=infinity)=limit((1/n)*(product(n+k,k=1..n))^(1/n),n=infinity);

Limit((Product(k+n, k = 1 .. n))^(1/n)/n, n = infinity) = 4*exp(-1)