Limits and Infinites

Limits are a concept that goes back at least as far as Archimedes, a Greek scientist, engineer and mathematician in the 3rd century BCE. He calculated a value for π based on the limit of measurements external and internal to a circle.

Rules for limits

If ${lim}↙{n→+∞} a_n = a$, and ${lim}↙{n→+∞} b_n = b$

Then, (i) ${lim}↙{n→+∞} a_n ± b_n = a ± b$

(ii) ${lim}↙{n→+∞} a_n ⋅ b_n = a ⋅ b$

(iii) ${lim}↙{n→+∞} ({a_n}/{b_n}) = a/b$

If ${lim}↙{n→+∞} a_n = a$, ⇒ ${lim}↙{n→+∞} 1/{a_n} = 1/a$

If ${lim}↙{n→+∞} a_n = +∞$, ${lim}↙{n→+∞} b_n = +∞$ ⇒ ${lim}↙{n→+∞} (a_n + b_n) = +∞$

If ${lim}↙{n→+∞} a_n = +∞$, ${lim}↙{n→+∞} b_n = +∞$ ⇒ ${lim}↙{n→+∞} (a_n - b_n) = ∞ - ∞$: i.e. indeterminate solution

${lim}↙{n→+∞} 1/{n^a} = 0$ (a > 0)

${lim}↙{n→+∞} a^n = 0$ (|a| < 1)

${lim}↙{n→+∞} ^n√{a} = 1$ (a > 0)

${lim}↙{n→+∞} ^n√{n} = 1$

${lim}↙{n→+∞} {(log n)^b}/{n^a} = 0$ (a > 0, b ∈ ℝ)

${lim}↙{n→+∞} {n^b}/{a^n} = 0$ (b > 0, |a| > 1)

${lim}↙{n→+∞} {a^n}/{n!} = 0$ (a ∈ ℝ)

${lim}↙{n→+∞} {n!}/{n^n} = 0$

${lim}↙{n→+∞} (1 + 1/n)^n = e$

${lim}↙{n→+∞} (1 + a/n)^{b⋅n} = e^{a⋅b}$, (a, b ∈ ℝ)

${lim}↙{x→±∞} (f(x) ± g(x)) = {lim}↙{x→±∞} f(x) ± {lim}↙{x→±∞} g(x) = L_1 ± L_2$

${lim}↙{x→±∞} (f(x) ⋅ g(x)) = {lim}↙{x→±∞} f(x) ⋅ {lim}↙{x→±∞} g(x) = L_1 ⋅ L_2$

${lim}↙{x→±∞} (f(x) ÷ g(x)) = {lim}↙{x→±∞} f(x) ÷ {lim}↙{x→±∞} g(x) = L_1 ÷ L_2$, where $L_2 ≠ 2$

${lim}↙{x→±∞} kf(x) = k {lim}↙{x→±∞} f(x) = kL_1$

${lim}↙{x→±∞} [f(x)]^{a/b} = L_1^{a/b}, a/b ∈ ℚ$, provided $L_1^{a/b} $ is real

Convergence of a Series

The sum of a finite geometric series is: $S_n = {u_1(1-r^n)}/{1-r}$, where $r$ is the common ratio of two consecutive terms, and $n$ is the number of terms $u$.

For a geometric series, ${Σ}↙{n=0}↖{∞} = {lim}↙{n→∞} {u_1(1-r^n)}/{1-r}$.

When $-1 < r < 1$, ${lim}↙{n→∞} r^n = 0$, and the series converges to its sum, $S={u_1}/{1-r}$.

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