Question

Prove that
Sup(n/2n+1, n=1,2,3.....)=1/2
I need solutions ASAP

Answer 1

Given:

A sequence n/ 2n + 1 , n = 1, 2,3 ..

To Prove:

that  Sup(n/2n+1, n=1,2,3.....) = 1/2

Solution:

We know n/2n + 1 = 0.5 x (2n + 1 - 1 )/(2n + 1)

n/2n + 1 = 1/2 ( 1 - 1/(2n+1)

• Since 1 > 1 − 1/2n+1 for each n ∈ N, 1 is an upper bound of the set.

Moreover,

• for each e > 0, by Archimedean Property, there exists N ∈ N|
• such that 2N+1  ≥  1/e.
• Then, 1 − 1/(2N+ 1) > 1 − e and therefore 1 − e is not an upper bound of the set for  each e > 0.

Therefore

• We conclude that 1 is the supremum of the set {1 − 1/2n+1 : n ∈ N}.

Therefore 1/2 x ( 1 - 1/2n+1 ) will have a supremum of 1/2.

Other Solution:

lim n-> ∞ n/2n+1 = lim n->∞ 1/( 2 + 1/n) = 1/2

Therefore value converges to 1/2.

Thus proved that the sup(n/2n+1) = 1/2.

Answer 2

Answer:

Thus proved that the sup(n/2n+1) = 1/2.p-by-step explanation: