Use the sandwich lemma to solve --> xn := 1/ sqrt (n^ 2 +1) + 1 /sqrt (n^ 2 +2) +...+ 1 /sqrt (n^ 2 +n) converges to 1

Answer 1

Write S_n for the series in question. Let a_n=n/sqrt(n^2+n), and b_n=n/sqrt(n^2+1). Then it is clear that a_n<S_n<b_n. Since both a_n and b_n converge to 1 as n approaches infinity, so does S_n by the Squeezing Principle or Sandwich Lemma as it is named in the question.