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I saw someone interpret $\sum_{i=1}^{n}\mathcal{O}\left(i^{k-2}\right)$ as $\mathcal{O}\left(n^{k-1}\right)$. Is this right? If so, can you explain?

Black Milk
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    Think for small values; e.g., with $k=2, 3$ and $n=1,2,3$. This should give a clue. –  Nov 01 '13 at 11:10
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    Could you clarify your question: $ \mathcal{O}(i^{k-2})$ with respect to ...? – user64494 Nov 01 '13 at 11:22
  • ...and observe that $\mathcal O$ is kinda "linear", i.e. $\sum \mathcal O(...) = \mathcal O(\sum ...)$ – AndreasT Nov 01 '13 at 11:22
  • @user64494 are you suggesting that I change the title of my query? – Black Milk Nov 01 '13 at 11:25
  • @AndreasT thank you, that linear aspect really helped. Where does this "linearity" of big O follow from? – Black Milk Nov 01 '13 at 11:33
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    Did you mean $\mathcal O(n^{k-1})$? – Hagen von Eitzen Nov 01 '13 at 11:50
  • @HagenvonEitzen I mean what I originally posted. This interpretation came from an answer to the following post: Asymptotic Behaviour of summation involving consecutive integer powers – Black Milk Nov 01 '13 at 11:54
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    The linearity is inherited by that of the limits. $f(n)\in\mathcal O(\alpha) + \mathcal O(\beta)$ means that $f=f_\alpha + f_\beta$ for some $f_\alpha\in\mathcal O(\alpha)$, $f_\beta\in\mathcal O(\beta)$; assuming wlog $\mathcal O(\alpha)\subseteq\mathcal O(\beta)$ $$ \lim_{n\to\infty}\frac{f}\beta = \lim_{n\to\infty}\frac{f_\alpha}\beta + \lim_{n\to\infty}\frac{f_\beta}\beta \in\mathbb R $$ therefore $f\in\mathcal O(\beta) = \mathcal O(\alpha+\beta)$. This proves that $\mathcal O(\alpha)+\mathcal(\beta)\subseteq\mathcal O(\alpha+\beta)$. A similar reasoning proves the other inclusion. – AndreasT Nov 01 '13 at 12:31
  • @AndreasT Thanks, I believe you meant to conclude that $\mathcal O(\alpha)+\mathcal O(\beta)\subseteq\mathcal O(\alpha+\beta)$ – Black Milk Nov 01 '13 at 13:02

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This is not quite true in this simplicity or at least depends on interpretation and context. One might interprete the statement as follows: Suppose that for each $n\in\mathbb N$ we have a function $f_n\colon \mathbb N\to\mathbb R$. Then we can define $f\colon \mathbb N\to \mathbb R$, $n\mapsto\sum_{i=1}^n f_i(n)$. Now if $f_n(x)=O(x^k)$ as $x\to \infty$ it does not follow that $f(x)=O(x^{k+1})$ as $x\to\infty$. For example, let $$f_i(n)=\begin{cases}n!&\text{if }n=i\\0&\text{otherwise}.\end{cases}$$ Then $f_i(n)=O(1)$ as $n\to\infty$ and $f(n)=n!$.

On the other hand, assume we have $g\colon\mathbb N\to\mathbb R$ and define $f\colon \mathbb N\to \mathbb R$, $n\mapsto\sum_{i=1}^n g(i)$. Now if $g(x)=O(x^k)$ as $x\to\infty$ we have $f(x)=O(x^{k+1})$ as $x\to\infty$. Indeed, assume $|g(x)|<cx^k$ for $x>N$. Then with $A=\left|\sum_{i=1}^Ng(i)\right|$ we have $$|f(x)|<A+\sum_{i=N+1}^n |g(i)|<A+c\sum_{i=N+1}^ni^k\le A+c(n-N)n^k<(c+A)n^{k+1}$$ for $n>N$.

Hagen von Eitzen
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