Limit of a Sequence

Convergent and divergent sequences

Consider these sequences $$ 1,\frac{1}{2},\frac{1}{3},\cdots,\frac{1}{n},\cdots $$ $$ 1,3,5,7,\cdots,2n-1,\cdots $$ $$ 5,3,1,-1,\cdots,-2n+7,\cdots $$ $$ -1,1,-1,\cdots,(-1)^n,\cdots $$ $$ -1,2,-3,\cdots,(-1)^nn,\cdots $$ As \(n\) gets larger and larger, let's see how the \(n\)th term changes.
$$ a_{n}=\frac{1}{n} $$ As \(n\) approaches infinity, You can see that the sequence \(a_{n}\) gets closer to \(0\)

In a sequence \(\{a_{n}\}\), as \(n\) approaches infinity, if \(a_{n}\) get closer to a certain value \(\alpha\), we call the the sequence \(\{a_{n}\}\) convergent, and we call \(\alpha\) the limit of the sequence \(\{a_{n}\}\). We represent it like this: $$ \lim_{n \rightarrow \infty}a_{n}=\alpha $$ Note 1 In \( a_{n}=\frac{1}{n} \), \(\lim_{n \rightarrow \infty}a_{n}=\lim_{n \rightarrow \infty}\frac{1}{n}=0\)
Note 2 In a sequence \(\{a_{n}\}\) \(\alpha, \alpha, \cdots, \alpha, \cdots\), we also say this sequence converges to \(\alpha\), and we can represent it like this:$$\lim_{n \rightarrow \infty}a_{n}=\lim_{n \rightarrow \infty}\alpha=\alpha $$


$$ b_{n}=2n-1 $$ As \(n\) approaches infinity, You can see that the sequence \(b_{n}\) also approaches infinity. $$ c_{n}=-2n+7 $$ As \(n\) approaches infinity, You can see that absolute value of the sequence \(c_{n}\) approaches infinity.
We call these sequences divergent.
We can represent the limit of these sequences like this: $$ \lim_{n \rightarrow \infty}b_{n}=\infty $$ $$ \lim_{n \rightarrow \infty}c_{n}=-\infty $$
$$ d_{n}=(-1)^n $$ $$ e_{n}=(-1)^nn $$ In these sequences, as \(n\) approaches infinity, the \(n\)th term neither converges nor approaches infinity (or negative infinity). Therefore, we call it divergent.

Note 3 We cannot write like this: $$ \lim_{n \rightarrow \infty}(-1)^n=\pm{1} $$

Properties

For any converging sequences \(\{a_{n}\},\{b_{n}\}\), let \(\lim_{n \rightarrow \infty}a_{n}=\alpha, \lim_{n \rightarrow \infty}b_{n}=\beta\) $$ \lim_{n \rightarrow \infty}ka_{n}=k\alpha $$ $$ \lim_{n \rightarrow \infty}(a_{n}\pm b_{n})=\alpha \pm\beta $$ $$ \lim_{n \rightarrow \infty}a_{n}b_{n}=\alpha\beta $$ $$ \lim_{n \rightarrow \infty}\frac{a_{n}}{b_{n}}=\frac{\alpha}{\beta} (b_{n}\ne 0, \beta\ne 0) $$ Keep in mind that these properties only work for converging sequences \(\{a_{n}\},\{b_{n}\}\)


Example

1. Evaluate the following
a) \(\lim_{n \rightarrow \infty}\frac{5}{n}=\lim_{n \rightarrow \infty}\left(5\times \frac{1}{n}\right)=5\times 0=0\)
b) \(\lim_{n \rightarrow \infty}\frac{n+1}{n}=\lim_{n \rightarrow \infty}\left(1+ \frac{1}{n}\right)=1+0=1\)
c) \(\lim_{n \rightarrow \infty}\frac{n^2}{2n}=\lim_{n \rightarrow \infty}\frac{n}{2}=\infty\)

Use the fact that \(\lim_{n \rightarrow \infty}\frac{1}{n}=0\)


Practice

1. Evaluate the following
a) \(\lim_{n \rightarrow \infty}\frac{2n^2-3n}{3n^2+2n-1}\)
b) \(\lim_{n \rightarrow \infty}\frac{n^2+3n}{2n-1}\)
c) \(\lim_{n \rightarrow \infty}\frac{n^2+1}{n^3-2n}\)
d) \(\lim_{n \rightarrow \infty}\left\{\log(n+2)-\log(n)\right\}\)

2. Evaluate the following
a) \(\lim_{n \rightarrow \infty}\frac{1^2+2^2+3^3+\cdots+n^2}{n^3+2}\)
b) \(\lim_{n \rightarrow \infty}\frac{1\times 2+2\times 3+3\times 4+\cdots+n(n+1)}{n(1+2+3+\cdots+n)}\)

Limit of a geometric sequence

Convergence and divergence of geometric sequence $$ r, r^2, r^3, \cdots, r^{n-1}, r^n, \cdots $$ depends on what value \(r\) is.

Let's examine \(\lim_{n \rightarrow \infty}r^n\)
If \(r\gt 1\), \(\lim_{n \rightarrow \infty}r^n=\infty\)
If \(r=1\), \(\lim_{n \rightarrow \infty}r^n=1\)
If \(|r|\lt 1 \), \(\lim_{n \rightarrow \infty}r^n=0\)
If \(r=-1\), \(\lim_{n \rightarrow \infty}r^n\) does not exist
If \(r\lt-1\), \(\lim_{n \rightarrow \infty}r^n\) does not exist