If it converges, find the limit

Question

Determine whether the sequence diverges or converges √(7n+11)-√(7n) converges or diverges. If it converges, find the limit.

Answer 1

a_n = √(7n+11) - √(7n) 

 

Multiply by the conjugate [√(7n+11) + √(7n)] / [√(7n+11) + √(7n)] to get 

 

a_n = [(7n+11) - 7n] / [√(7n+11) + √(7n)] = 11 / [√(7n+11) + √(7n)]

 

As n-->∞ the numerator is a constant while the denominator is unbounded, so 

 

lim(n-->∞) a_n = 0  ==>  the sequence converges to 0.