Elementary Calculus: Theorem 1: Sequences approaching ∞

THEOREM 1

Each of the following sequences approaches ∞.

lim_n→∞ n! = ∞,

lim_n→∞ bⁿ = ∞ if b > 1,

lim_n→∞ n^c = ∞ if c > 0,

lim_n→∞ ln(n) = ∞.

Moreover, each sequence in the list grows faster than the next one,

(i) (b > 1),

(ii) (b > 1, c> 0),

(iii) (c > 0).

PROOF

Let H be positive infinite. We already know that ln H is positive infinite. We must show that each of the following are also positive infinite.

It is easier to show that their logarithms are positive infinite. We need the fact that, by l'Hospital's rule for ∞/∞,

so for all infinite K.

(i) . Let m > b. Then

= ln 1 + ... + ln (m - 1) + ln m + ... + ln H - H ln b

> (H - m) ln m - H ln b = H(ln m - ln b) - m ln m. Since m > b. ln m > ln b, and ln (H!/b^H) is positive infinite.

(ii)

Since b > 1, ln b > 0. (ln H)/H is infinitesimal. Therefore ln(b^H/H^c) is

positive infinite, (iii) . Let K = ln H.

Since c > 0, K is infinite, and (ln K)/K is infinitesimal, ln(H^c/ln H)) is positive infinite.

Note: For n = 1, the term n^c/(ln n) is undefined, so we should start the sequence with n = 2.