What is sequence? Is the sequence

\(a_{n} = \frac{n}{\sqrt{5 + n}}\)

convergent?

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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

 

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