CREDIT - ADITYA BHOJ
ALGEBRA, INTEGRAL, INEQUALITIES .
PROBLEM- 01
If we define the function for n ≥ 1
n
X 1 πm 1 πm
ϕ(n) = sin (−1) 2m(m+1) sin (−1) 2m(m−1)
m=1
4 2
then prove that:
∞
X n 3 √
2− 2 ϕ(n) = 2+ 2
n=1
5
PROBLEM- 02
Z sin−1 (1−ε)
cos x
Ω= lim log (cos x)cot x · (sin x) 1+sin x dx
ε→0, ε>0 sin−1 ε
PROBLEM -03
If we have the integral relation
!2 !2
∞ 1 1 ∞ 1 1
sin x − cos x + sin x + cos x −
Z Z
x x 3πβ x x
√ dx = + √ dx
−∞
3
x 4 −∞
3
x
then prove that β 4 + 3β 2 + 9 = 0
PROBLEM -04
a, b, c – sides in △ABC, u, c, w ≥ 0, xn , yn , zn > 0, n ∈ N, n ≥ 1
xn+1 yn+1 zn+1
lim = a, lim = b, lim =c
n→∞ nxn n→∞ nyn n→∞ nzn
Prove that:
√ √ √ √
aun xn bv n yn cwn zn
4S uv + vw + wu
lim + + ≥
n→∞ n n n e
1
ALGEBRA, INTEGRAL, INEQUALITIES .
PROBLEM- 01
If we define the function for n ≥ 1
n
X 1 πm 1 πm
ϕ(n) = sin (−1) 2m(m+1) sin (−1) 2m(m−1)
m=1
4 2
then prove that:
∞
X n 3 √
2− 2 ϕ(n) = 2+ 2
n=1
5
PROBLEM- 02
Z sin−1 (1−ε)
cos x
Ω= lim log (cos x)cot x · (sin x) 1+sin x dx
ε→0, ε>0 sin−1 ε
PROBLEM -03
If we have the integral relation
!2 !2
∞ 1 1 ∞ 1 1
sin x − cos x + sin x + cos x −
Z Z
x x 3πβ x x
√ dx = + √ dx
−∞
3
x 4 −∞
3
x
then prove that β 4 + 3β 2 + 9 = 0
PROBLEM -04
a, b, c – sides in △ABC, u, c, w ≥ 0, xn , yn , zn > 0, n ∈ N, n ≥ 1
xn+1 yn+1 zn+1
lim = a, lim = b, lim =c
n→∞ nxn n→∞ nyn n→∞ nzn
Prove that:
√ √ √ √
aun xn bv n yn cwn zn
4S uv + vw + wu
lim + + ≥
n→∞ n n n e
1
