# Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.06 Limit Laws Example

## Example

Determine whether the sequence

{\displaystyle {\begin{aligned}a_{n}={\frac {n^{2}}{2n^{2}+9}}\end{aligned}}}

converges.

## Complete Solution

Using the limit laws for infinite sequence, we would evaluate

{\displaystyle {\begin{aligned}\lim _{n\rightarrow \infty }a_{n}&=\lim _{n\rightarrow \infty }{\frac {n^{2}}{2n^{2}+9}}&&(1)\\\\&=\lim _{n\rightarrow \infty }{\frac {1}{2+{\frac {9}{n^{2}}}}}&&(2)\\\\&={\frac {\lim _{n\rightarrow \infty }1}{\lim _{n\rightarrow \infty }2+9\cdot \lim _{n\rightarrow \infty }{\frac {1}{n^{2}}}}}&&(3)\\\\&={\frac {1}{2+0}}&&(4)\\\\&={\frac {1}{2}}&&(5)\end{aligned}}}

Because our limit evaluates to a finite number, the sequence converges (and it converges to 1/2).

The above example is trivial, but demonstrates why we need the limit laws. They allow us to evaluate limits of more complicated sequences.

## Explanation of Each Step

### Step (1)

We applied the definition of convergence of a sequence. Recall that to determine if a sequence is convergent we evaluate

{\displaystyle {\begin{aligned}\lim _{n\rightarrow \infty }a_{n}\end{aligned}}}

and if this limit exists, the sequence converges. If it doesn't the sequence is divergent.

### Step (2)

To make the limit easier to evaluate, we divided both the numerator and denominator by ${\displaystyle n^{2}}$. This is as commonly used trick when evaluating limits.

### Step (3)

{\displaystyle {\begin{aligned}\lim _{n\rightarrow \infty }a_{n}&=\lim _{n\rightarrow \infty }{\frac {1}{2+{\frac {1}{n^{2}}}}}&&(2)\\\\&={\frac {\lim _{n\rightarrow \infty }1}{\lim _{n\rightarrow \infty }{\Big (}2+{\frac {9}{n^{2}}}{\Big )}}}&&\mathrm {law\ 4)} \\\\&={\frac {\lim _{n\rightarrow \infty }1}{\lim _{n\rightarrow \infty }(2)+\lim _{n\rightarrow \infty }{\Big (}{\frac {9}{n^{2}}}{\Big )}}}&&\mathrm {law\ 1)} \\\\&={\frac {\lim _{n\rightarrow \infty }1}{\lim _{n\rightarrow \infty }(2)+9\cdot \lim _{n\rightarrow \infty }{\Big (}{\frac {1}{n^{2}}}{\Big )}}}&&\mathrm {law\ 2)} \\\\\end{aligned}}}

### Steps (4, 5)

These two steps consisted of evaluating our limits and simple algebraic manipulation and should be straightforward to students who are familiar with limits of functions.

## Possible Challenges

### Memorizing the Limit Laws

Students are expected to memorize the laws, but not the labels we assigned to them. For example, students are expected to have memorized that

${\displaystyle \lim _{n\rightarrow \infty }a_{n}b_{n}=\lim _{n\rightarrow \infty }a_{n}\cdot \lim _{n\rightarrow \infty }b_{n}}$

but do not need to memorize that this is the third law.