# What is sequence? Is the sequence$$a_{n} = \frac{n}{\sqrt{5 + n}}$$convergent?

A sequence is an ordered list of things. Such thinks may finite or infinite. In a sequence, the terms are separated by commas(,).

Definition:

A sequence is a list of numbers which are written as a definite order:

a1, a2, a3, a4, …….. an

Here, a1 is first term, a2 is second term and likewise an is the nth term of the series.

For example:

{1, 2, 3, 4, 5 ……} is a sequence

Problem Part:

The general form of the series is

$$a_{n} = \frac{n}{\sqrt{5 + n}}$$

Using D’Alembert’s Ratio Test,

$$\lim_{n \to ∞} a_n = \lim_{n \to ∞} \frac{n}{\sqrt{5 + n}}$$

$$= \lim_{n \to ∞} \frac{1}{ \frac{1}{n} \sqrt{5 + n}}$$

$$= \lim_{n \to ∞} \frac{1}{ \sqrt{\frac{5}{n^2} + \frac{1}{n}}}$$

$$= \lim_{n \to ∞} \frac{1}{ \sqrt{\frac{5}{∞} + \frac{1}{∞}}}$$

$$= \frac{1}{∞}$$

= 0

So, The given sequence is convergent