Mattie Monroe

2022-10-13

what the following may equal?

$\sum _{i=1}^{n}i\lceil {\mathrm{log}}_{2}(i)\rceil =?$

$\sum _{i=1}^{n}i\lceil {\mathrm{log}}_{2}(i)\rceil =?$

faux0101d

Beginner2022-10-14Added 21 answers

Use sum by parts:

$\sum _{1\le k\le n}{x}_{k}\mathrm{\Delta}{y}_{k}={x}_{n+1}{y}_{n+1}-{x}_{1}{y}_{1}-\sum _{1\le k\le n}{y}_{k+1}\mathrm{\Delta}{x}_{k}$

In your case $\mathrm{\Delta}{y}_{k}=\lceil {\mathrm{log}}_{2}k\rceil $ and ${x}_{k}=k$. You'll get a sum similar to the original, you should be able to solve for that one.

$\sum _{1\le k\le n}{x}_{k}\mathrm{\Delta}{y}_{k}={x}_{n+1}{y}_{n+1}-{x}_{1}{y}_{1}-\sum _{1\le k\le n}{y}_{k+1}\mathrm{\Delta}{x}_{k}$

In your case $\mathrm{\Delta}{y}_{k}=\lceil {\mathrm{log}}_{2}k\rceil $ and ${x}_{k}=k$. You'll get a sum similar to the original, you should be able to solve for that one.

caschaillo7

Beginner2022-10-15Added 1 answers

I could come with a solution with an incisive precision for any base a by doing this:

$\phantom{\rule{0ex}{0ex}}T(n)=\sum _{i=1}^{n}i\lceil {\mathrm{log}}_{a}(i)\rceil =\left[\sum _{i=1}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}i\left(\sum _{j={a}^{i-1}+1}^{{a}^{i}}j\right)\right]+\lceil {\mathrm{log}}_{a}(n)\rceil \sum _{i={a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+1}^{n}i\phantom{\rule{0ex}{0ex}}T(n)=\left[\frac{1}{2}\sum _{i=1}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}i({a}^{i-2}(a-1)({a}^{i+1}+{a}^{i}+a))\right]+\lceil {\mathrm{log}}_{a}(n)\rceil \sum _{i={a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+1}^{n}i\phantom{\rule{0ex}{0ex}}\text{Let}{T}_{1}(n)=\left[\frac{1}{2}\sum _{i=1}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}i({a}^{i-2}(a-1)({a}^{i+1}+{a}^{i}+a))\right]\phantom{\rule{0ex}{0ex}}\text{And}{T}_{2}(n)=\lceil {\mathrm{log}}_{a}(n)\rceil \sum _{i={a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+1}^{n}i\phantom{\rule{0ex}{0ex}}{T}_{1}(n)=\frac{1}{2({a}^{2}-1)}((-1-\lfloor {\mathrm{log}}_{a}(n)\rfloor ){a}^{2\lfloor {\mathrm{log}}_{a}(n)\rfloor}-{a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor +1}+\lfloor {\mathrm{log}}_{a}(n)\rfloor {a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor +2}+\lfloor {\mathrm{log}}_{a}(n)\rfloor {a}^{2\lfloor {\mathrm{log}}_{a}(n)\rfloor +2}-(\lfloor {\mathrm{log}}_{a}(n)\rfloor +1){a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+a+2)\phantom{\rule{0ex}{0ex}}{T}_{2}(n)=\frac{\lceil {\mathrm{log}}_{a}(n)\rceil (n-{a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor})({a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+n+1)}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Therefore}T(n)={T}_{1}(n)+{T}_{2}(n)$

$\phantom{\rule{0ex}{0ex}}T(n)=\sum _{i=1}^{n}i\lceil {\mathrm{log}}_{a}(i)\rceil =\left[\sum _{i=1}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}i\left(\sum _{j={a}^{i-1}+1}^{{a}^{i}}j\right)\right]+\lceil {\mathrm{log}}_{a}(n)\rceil \sum _{i={a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+1}^{n}i\phantom{\rule{0ex}{0ex}}T(n)=\left[\frac{1}{2}\sum _{i=1}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}i({a}^{i-2}(a-1)({a}^{i+1}+{a}^{i}+a))\right]+\lceil {\mathrm{log}}_{a}(n)\rceil \sum _{i={a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+1}^{n}i\phantom{\rule{0ex}{0ex}}\text{Let}{T}_{1}(n)=\left[\frac{1}{2}\sum _{i=1}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}i({a}^{i-2}(a-1)({a}^{i+1}+{a}^{i}+a))\right]\phantom{\rule{0ex}{0ex}}\text{And}{T}_{2}(n)=\lceil {\mathrm{log}}_{a}(n)\rceil \sum _{i={a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+1}^{n}i\phantom{\rule{0ex}{0ex}}{T}_{1}(n)=\frac{1}{2({a}^{2}-1)}((-1-\lfloor {\mathrm{log}}_{a}(n)\rfloor ){a}^{2\lfloor {\mathrm{log}}_{a}(n)\rfloor}-{a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor +1}+\lfloor {\mathrm{log}}_{a}(n)\rfloor {a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor +2}+\lfloor {\mathrm{log}}_{a}(n)\rfloor {a}^{2\lfloor {\mathrm{log}}_{a}(n)\rfloor +2}-(\lfloor {\mathrm{log}}_{a}(n)\rfloor +1){a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+a+2)\phantom{\rule{0ex}{0ex}}{T}_{2}(n)=\frac{\lceil {\mathrm{log}}_{a}(n)\rceil (n-{a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor})({a}^{\lfloor {\mathrm{log}}_{a}(n)\rfloor}+n+1)}{2}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\text{Therefore}T(n)={T}_{1}(n)+{T}_{2}(n)$

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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