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Test the convergence of ∑︁∞
n=1
[︂
n!2n
2
n
]︂
.
Ans: ∑︁un Converges
25. Test the convergence of ∑︁∞
n=1
[︂
n
3 + 1
2
n + 1]︂
.
Ans: ∑︁un Converges
Hint: Use vn as
1
√n
.
26. Test the convergence of
1
2 · 3 · 4
+
2
3 · 4 · 5
+
3
4 · 5 · 6
+
4
5 · 6 · 7
+ · · · .
Ans: ∑︁un Converges
Hint: Use vn as
1
n2
.
27. Test the convergence of
√
2 − 1
3
3 − 1
+
√
3 − 1
4
3 − 1
+
√
4 − 1
5
3 − 1
+ · · · .
Ans: ∑︁un Converges
Hint: Use vn as
1
n
5
2
.
28. Test the convergence of
2
1
p +
3
2
p +
4
3
p + · · · .
Ans: ∑︁un Converges if p > 2 and ∑︁un Diverges if p ≤ 2
Hint: Use vn as
1
n
p−1
.
9. Find the expression for ∫︁∫︁∫︁∫︁
V
x
l−1y
m−1z
n−1 dx by DZ(Dirichlet’s Integral) in
the form of Gamma integrals, here V is the region x ≥ 0, y ≥ 0, z ≥ 0 and
x + y + z ≤ a.
Hint/Ans: Γ(l)Γ(m)Γ(n)
Γ(l + m + n + 1)
a
l+m+n
10. Evaluate ∫︁
1
1
√
1 − x4
dx
A. A company calculates its expected profits as a function of the quantity of the items it can sell. How much are their expected profits if the company’s profit function is P(q)=q³−2000q+500 and their current sales quantity, q, is 60?
[DM] Define group. Show that the set P3 of all permutations on three symbols 1,2,3 is a finite non-abelian group of order six with respect to permutation multiplication as composition.
The marketing department of Spager Ltd estimated that if the selling price of product is set at $15 per unit then the sales will be 50 units per week, while, if the selling price is set at $20 per unit, the sales will be 30 units per week. Assume that the graph of this function is linear. The production department estimates that the variable cost will be $5 per unit and that the fixed cost will be $50 per week, and special cost are estimated as $0.125x2, where x is the quantity of output. All production is sold.
(a) Show that the relationship between price (Pr) and quantity sold (x) , are given by the equation Pr = 27.5 - 0.25x.
(b) Find the revenue function, R.
(c) Find the total cost function (C).
(d) Advise the company on production and pricing policy if it wishes to maximize profits, and find the maximum profit.
[C] Find the Wronskian of the following functions and determine whether it is linearly dependent or linearly independent on (-∞,∞).
- {x2, x+1, x-3} ans, W=8, linearly independent
- {3e2x, e2x} ans, W=0, linearly dependent
- {x2, x3, x4} ans, W=2x^6, linearly independent
use Stokes theorem evaluate ∮A.dr, where A=-5yi+4xj+zk and C is cirvle x²+y²=4,z=1
using Greens theorem in the plane evaluate ∮(3x+4y)dx+(2x-3y)dy where C is the circle x²+y²=4 traversed in the counterclockwise sense
If fie=2x+2, evaluate volume integral where R is the region of the cube bounded by the plane x=0 ,x=1, y=0, y=1 ,z=0, z=1
Determine the length of the curve 𝑥 = 𝑦^2 /2 for 0 ≤ 𝑥 ≤ 1/2 . Assume 𝑦 positive.
