Question

How to prove that $2^n\not\in O(n^k)$ through proof by contradiction

score 112 · Answer 1 · 2026-03-17 22:48:19

112

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How to prove that $2^n\not\in O(n^k)$ through proof by contradiction

Published on 17 Mar 2026 - 22:48

score 54 · Answer 2 · 2026-03-17 08:32:45

54

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Does converge $\sum_{n=2}^\infty\frac{1}{\varphi(p_n-2)-1+p_n}$, where $\varphi(n)$ is the Euler's totient function and $p_n$ the $n$th prime number?

Published on 17 Mar 2026 - 8:32

score 107 · Answer 3 · 2026-03-17 18:38:17

107

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On the behaviour of $\frac{1}{N}\sum_{k=1}^N\frac{\pi(\varphi(k)+N)}{\varphi(\pi(k)+N)}$ as $N\to\infty$

Published on 17 Mar 2026 - 18:38

score 682 · Answer 4 · 2026-03-22 04:34:31

682

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Show that if f(n) is O(g(n) and d(n) is O(h(n)), then f(n) + d(n) is O(g(n) + h(n))

Published on 22 Mar 2026 - 4:34

score 28 · Answer 5 · 2018-01-19 12:51:55

28

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An easy way to argue $((n+1)\log_2(n+1)-n) = \mathcal{O}(n*\log(n))$

Published on 19 Jan 2018 - 12:51

score 43 · Answer 6 · 2018-01-19 17:45:14

43

Views

How to formally to show one term is lower order than the other terms

Published on 19 Jan 2018 - 17:45

score 34.2k · Answer 7 · 2026-03-22 05:18:30

34.2k

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Show that $2^{n+1}$ is $O(2^n)$

Published on 22 Mar 2026 - 5:18

score 21 · Answer 8 · 2026-03-17 21:16:19

21

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Solving $T(n)=T(n-1)+c\cdot \log n,\quad T(1)=d$

Published on 17 Mar 2026 - 21:16

score 44 · Answer 9 · 2026-03-16 19:39:39

44

Views

Big - O proof help

Published on 16 Mar 2026 - 19:39

score 141 · Answer 10 · 2026-03-22 07:53:18

141

Views

The behaviour of $\sum_{\mathcal{R}\leq x}\left\{\frac{x}{\mathcal{R}}\right\}$, where $\mathcal{R}$ denote Ramanujan primes

Published on 22 Mar 2026 - 7:53

score 54 · Answer 11 · 2026-03-17 06:40:45

54

Views

Verify asymptotics as $n\to\infty$

Published on 17 Mar 2026 - 6:40

score 321 · Answer 12 · 2026-03-22 07:57:13

321

Views

How to understand intuitively the fact that $\log(n!) = n\log(n) - n + O(\log(n))$?

Published on 22 Mar 2026 - 7:57

score 2.6k · Answer 13 · 2026-03-17 02:49:50

2.6k

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Big O and Small o notation (Taylor series)

Published on 17 Mar 2026 - 2:49

score 28 · Answer 14 · 2026-03-16 23:14:50

28

Views

Some asymptotics with log

Published on 16 Mar 2026 - 23:14

score 50 · Answer 15 · 2026-03-16 23:20:10

50

Views

Show $\lim_n f(n)=0$

Published on 16 Mar 2026 - 23:20

List Question

How to prove that $2^n\not\in O(n^k)$ through proof by contradiction

Does converge $\sum_{n=2}^\infty\frac{1}{\varphi(p_n-2)-1+p_n}$, where $\varphi(n)$ is the Euler's totient function and $p_n$ the $n$th prime number?

On the behaviour of $\frac{1}{N}\sum_{k=1}^N\frac{\pi(\varphi(k)+N)}{\varphi(\pi(k)+N)}$ as $N\to\infty$

Show that if f(n) is O(g(n) and d(n) is O(h(n)), then f(n) + d(n) is O(g(n) + h(n))

An easy way to argue $((n+1)\log_2(n+1)-n) = \mathcal{O}(n*\log(n))$

How to formally to show one term is lower order than the other terms

Show that $2^{n+1}$ is $O(2^n)$

Solving $T(n)=T(n-1)+c\cdot \log n,\quad T(1)=d$

Big - O proof help

The behaviour of $\sum_{\mathcal{R}\leq x}\left\{\frac{x}{\mathcal{R}}\right\}$, where $\mathcal{R}$ denote Ramanujan primes

Verify asymptotics as $n\to\infty$

How to understand intuitively the fact that $\log(n!) = n\log(n) - n + O(\log(n))$?

Big O and Small o notation (Taylor series)

Some asymptotics with log

Show $\lim_n f(n)=0$

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