Show the sequence defined by: Sn = √(n+1) –√n ∀ n ∈ N is convergent.

1. Show the sequence<Sn> defined by:

Sn = √(n+1) –√n ∀ n ∈ N is convergent.

[Note: ∀ = सभी के लिए]

[Note: ∈ = सदस्य हैं]

Sol: we have Sn = √(n+1) –√n

Sn = {√(n+1) –√n} × {√(n+1) +√n} ÷ {√(n+1) +√n}

Sn = 1 / √(n+1) +√n < 1/√n+√n = 

1/√2n < 1/√n

i.e., Sn<1/√n,

Let ∈ > 0, then | Sn – 0 | < 1/√n < ∈ provided √n> 1/∈ i.e. , n > 1/∈²

If m is a positive integer greater than 1/∈² then | Sn – 0 | < ∈ for all n≥m hence lim Sn= 0.

Thanks for watching......

Share and comment kare........