$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Next we discover what sequences such as $\displaystyle \frac{1}{n^2}$ and $\displaystyle \frac{1}{n^3}$ tend to. This will be very useful later when combined with other techniques. Theorem Let $p\in \N$. Then $\displaystyle \lim_{n\to \infty} n^{-p}\to 0$. Proof Suppose $\varepsilon >0$ is given. Then, $n>\varepsilon ^{-1/p}$ […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Theorem If $a_n\to a$, then $|a_n|\to |a|$. Proof Given $\varepsilon >0$, there exists $N$ such that $|a_n-a|\lt \varepsilon $ for all $n>N$. But by the reverse triangle inequality \[ \left| |a_n|-|a| \right| \leq |a_n-a| \lt \varepsilon {\text{ for all }} n>N. \] Hence, $|a_n|\to […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ What about sequences such as $1/2$, $1/4$, $1/8$, $1/16$, which can defined as $(1/2)^n$? The following proposition deals with this. It is very useful and is used countless times in mathematical proofs. The proof here uses properties of the log function, a function we […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Note that in the definition of convergence that the definite article is used for the limit, i.e., we have ‘the limit’ rather than ‘a limit’. A priori there is no reason to expect that a limit like that in the definition is unique. Well, […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Common mistake Probably the most common mistake about the definition of limit is to describe it as `the thing that the sequence gets closer to but never reaches’. I have seen this so many times. The first mistake, of course, is that this is […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Let’s now see some examples of sequences that do not converge, i.e., they are non-convergent sequences. Example The sequence $a_n=(-1)^n$ is not convergent. To show that $a_n$ does not have a limit we shall assume, for a contradiction, that it does. Let $a$ be […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Example Show that $a_n = \frac{3n^2}{n^2-5} $ has limit equal to $3$. We do the usual and consider $|a_n-a|$: \begin{align*} |a_n-3|&= \left| \frac{3n^2}{n^2-5} – 3 \right| \\ &= \left| \frac{3n^2-3n^2+15}{n^2-5} \right| \\ &= \left| \frac{15}{n^2-5} \right| \\ &= \frac{15}{\left| n^2-5 \right|} . \end{align*} Now […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ By using the triangle inequality we can simplify expressions when we have negative signs in the expression for the sequence. Example Show that $a_n=\frac{2n^4+5n^3-3n^2+1}{n^4} \to 2$. Again we simplify $|a_n-a|$ and find expressions that it is less than or equal to: \begin{align*} |a_n-2|&= \left| […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Let’s try a more complicated example. Example Show that $a_n=\frac{2n+3}{n^2}$ has limit $0$. We proceed as before by simplifying $|a_n-a|$ via equalities: \[ |a_n-0|=\left| \frac{2n+3}{n^2} – 0 \right| = \left| \frac{2n+3}{n^2} \right| = \frac{2n+3}{n^2} . \] Now what we want is to find $N$ […]

$$\def \C {\mathbb{C}}\def \R {\mathbb{R}}\def \N {\mathbb{N}}\def \Z {\mathbb{Z}}\def \Q {\mathbb{Q}}\def\defn#1{{\bf{#1}}}$$ Example Suppose that $a_n=c$ for all $n\in \N$. That is, $(a_n)$ is a constant sequence. Then, $\displaystyle \lim_{n\to \infty} a_n=c$. Proof Given any $\varepsilon$, let $N=1$. Then, for all $n>N$ we have \[ |a_n-c|=|c-c|=0\lt \varepsilon . \]