Monotonicalli Sequence Theorem

Question

Using the Monotonic Sequence Theorem prove that the following sequences are convergent. () =Σ 1/(+) () 1=1, +1=/√(2+1) , ∈ℤ+

Answer 1

In case (i) we have for all $n$,

$$S_n = \sum_{k=1}^n \frac{1}{n+k} < \sum_{k=1}^n \frac{1}{n+1} = \frac{n}{n+1} < 1.$$

Since each term is positive the sequence of partial sums is increasing and bounded above as shown -- hence, convergent.

In case (ii) we have

$$0 < S_{n+1} = \frac{S_n}{\sqrt{S_n^2+1}}< S_n.$$

In this case, the sequence is decreasing and bounded below.