6. 1-amaliy mashg’ulot Mavzu: Maple dasturidan matematik masalalarni yechishda foydalanish. Limitlarni hisoblash

amaliy mashg’ulot


Qator yig’indisi va ko’paytmasini hisoblash
Chekli va cheksiz yig’indi ni to’g’ridan-to’g’ri bajarish buyrug’i sum va bajarish bekor qilingan buyrug’i Sum orqali belgilanadi. Bu buyruqlarning parametrlari bir xil: Sum(t, n=a..b); va sum(t, n=a..b); bu yerda t – yig’indining indeksiga bog’liq bo’lgan ifoda, a..b – esa yig’indini n=a dan n=b gacha bajarilishini ko’rsatuvchi yig’indi indeksining chegarasi.
> sum('k^2', 'k'=0..4);

> sum('k^2', 'k'=0..n);

> sum('k^2', 'k');

> sum('a[k]*x^k','k'=0..4);

> sum('a[k]*x^k','k'=0..n);

> Sum('k/(k+1)','k'=0..n) = sum('k/(k+1)', 'k'=0..n);

> sum('k/(k+1)', 'k');

> sum('k*a^k', 'k');

Agar cheksiz qator yig’indisini hisoblash talab etilgan bo’lsa yuqori chegara sifatida infinity kiritiladi.
ko’paytma ham xuddi shunday bevosita bajarish buyrug’i product(P(n),n=a..b) va bajarilmaydigan buyrug’i Product P(n),n=a..b) yordamida belgilanadi
> product( k^2, k=1..4 );

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

> product( k^2, k );

> product( a[k], k=0..4 );

> product( a[k], k=0..n );

> Product( n+k, k=0..m ) = product( n+k, k=0..m );

> product( k, k=RootOf(x^3-2) );

Misollar
1.Umumiy hadi quyidagiga teng bo’lgan qatorning to’liq va N-qismi yig’indisini toping: a_n= .
> restart: a[n]:=1/((3*n-2)*(3*n+1));

> S[N]:=Sum(a[n], n=1..N)=sum(a[n], n=1..N);

> S:=limit(rhs(S[N]), N=+infinity);

2. Darajali qator qaysi funksiyaga yaqinlashadi: ?
> Sum((-1)^(n+1)*n^2*x^n, n=1..infinity)=sum((-1)^(n+1)*n^2*x^n, n=1..infinity);

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3. Darajali qator yig’indisini toping: .
> Sum((1+x)^n/((n+1)*n!), n=0..infinity)=sum((1+x)^n/((n+1)*n!), n=0..infinity);

4. Binomial qator yig’indisini toping: .
> Sum(binomial(n,4)*(1-x)^n, n=1..infinity)=sum(binomial(n,4)*(1-x)^n, n=1..infinity);

5. Cheksiz ko’paytmani hisoblang:
> Product((n^3-1)/(n^3+1),n=2..infinity)=product((n^3-1)/(n^3+1), n=2..infinity);
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6. Ifodani hisoblang:
>S