Blog

Parin Sir > Quantitative Aptitude (Maths) > Calculus > Limit of a Sequence > Limit of a Sequence

Limit of a Sequence

August 16, 2020
Category: Limit of a Sequence

No Comments

Limit of a Sequence

$\begin{matrix}lim\\n\rightarrow\infty\\\end{matrix} a_n = l$

⟺ ∀ε > 0, ∃ m ∈ N ∍

n ≥ m, n∈ N ⇒ $\left|a_n-l\ \right|$ < ε ; Where { $a_n$ } is a sequence.

$\begin{matrix}lim\\n\rightarrow\infty\\\end{matrix} \ \ \frac{1}{n} = 0$
$\begin{matrix}lim\\n\rightarrow\infty\\\end{matrix} \ r^n = 0$ ; |r| < 1

If { $S_n$ } is an infinite geometric series. i.e. = a + a r + a r² + . . . , then

$\begin{matrix}lim\\n\rightarrow\infty\\\end{matrix} S_n = \frac{a}{1-r}$

5 . $\begin{matrix}lim\\n\rightarrow\infty\\\end{matrix} \left(1+\frac{1}{n}\right)^n = e$ OR $\begin{matrix}lim\\x\rightarrow0\\\end{matrix} {(\ 1\ +\ x\ )}^\frac{1}{x} = e$

$\begin{matrix}lim\\h\rightarrow 0\\\end{matrix}$ $\frac{a^h-1}{h}$ = log a = log a OR $\begin{matrix}lim\\h\rightarrow 0\\\end{matrix}$ $\frac{e^h-1}{h}$ = 1
If a R⁺ and a > 1, then