All Questions
17
questions
0
votes
0
answers
43
views
Compute a tight upper bound of $\sum_{i=1}^{n-1}\frac{1}{3^i\log{n}- 3i}$?
I am trying to compute a tight upper bound of the sum below.
$\sum_{i=1}^{n-1}n\frac{\frac{1}{3^i}}{\log_3{(n/3^i)}}$
I was able to 'simplify' it up to the expression below.
$n\sum_{i=1}^{n-1}\frac{1}{...
- 185
0
votes
1
answer
48
views
How to simplify $\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}$
Is it possible to simplify
$$
S(\mathbf{x})=\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}
$$
A few observations:
$\sum_{k=0}^{n}\sum_{i=0}^k x_{i}=\sum_{k=0}^{n}(n+1-k)x_k$
$ e^{\...
- 3,435
2
votes
0
answers
39
views
Simplifying $\sum_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$ in two ways gives different results
I want to calculate the result of
$$\sum\limits_{k=1}^{\log_{2}{n}}\log_{3}{\frac{n}{2^k}}$$ I used two below approaches. Both approaches are based on $\log A + \log B = \log (A \times B)$ and $\sum\...
- 485
7
votes
1
answer
139
views
Trying to find a NICE form of : $\sum_{m=1}^{n}\lfloor\log_2m\rfloor$ [ Mathematical Gazette 2002 ]
$Q.$ Find a NICE form of : $$\sum_{m=1}^{p}\lfloor\log_2m\rfloor$$
APPROACH : We have , $$\lfloor\log_21\rfloor⠀\color{red}{\lfloor\log_22\rfloor}⠀\lfloor\log_23\rfloor⠀\color{red}{\lfloor\log_24\...
- 688
4
votes
4
answers
634
views
(AIME 1994) $ \lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \ldots + \lfloor \log_2 n \rfloor = 1994 $
$($AIME $1994)$ Find the positive integer $n$ for which
$$ \lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \ldots + \lfloor \log_2 n \rfloor = 1994 $$ where $\lfloor ...
- 41
0
votes
3
answers
2k
views
If $\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right)= \prod_{r = 10}^{99}\log _r(r+1)$, then find $n$.
If \begin{align}\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right) = \prod_{r = 10}^{99}\log _r(r+1).\end{align}
then find $n$.
I found this question in my 12th grade textbook and I just can't wrap ...
- 125
3
votes
0
answers
112
views
Evaluate $\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$ [duplicate]
Evaluate $$I=\int_{0}^{1} \frac{\ln(1-x)\ln^2(1+x)\:dx}{x}$$
We have $$\frac{\ln(1-x)}{x}=-\sum_{k=1}^{\infty}\frac{x^{k-1}}{k}$$
Hence $$I=-\sum_{k=1}^{\infty}\left(\frac{1}{k}\int_{0}^{1}x^{k-1}\...
- 10.2k
0
votes
1
answer
31
views
Likelihood ration Algebraic issue
I have got the following likelihood:
$$l(p) = C + xlog(p) + (n-x)log(1-p)$$
I have got that $\theta_0 = 1/3$ and $\theta_1 = 1/2$
All I need to do is find the correct value of the log likelihood ...
- 149
3
votes
4
answers
186
views
Summation or Integral representation ${e^{2}\above 1.5pt \ln(2)}=10.66015459\ldots$
How can I construct a summation or integral representation of $${e^2\above 1.5pt \ln(2)}.$$ Naively I would write $$\Bigg(\sum_{n=0}^{\infty}{2^n \above 1.5pt n!} \Bigg)\Bigg(\sum_{n=1}^\infty {(-1)^{...
- 3,758
6
votes
4
answers
828
views
Summing reciprocal logs of different bases
I recently took a math test that had the following problem:
$$
\frac{1}{\log_{2}50!} + \frac{1}{\log_{3}50!} + \frac{1}{\log_{4}50!} + \dots + \frac{1}{\log_{50}50!}
$$
The sum is equal to 1. I ...
- 171
1
vote
2
answers
180
views
Solving $x^{2n} = \frac{1}{2^n}$ for $x$
What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation
$$x^{2n} = \frac{1}{2^n}$$
for $...
- 11
1
vote
3
answers
927
views
$N =\sum_{k = 1}^{1000}k(\lceil\log_{\sqrt{2}}k\rceil-\lfloor\log_{\sqrt{2}}k\rfloor). $
Find $N$ for
$$N =\sum_{k = 1}^{1000}k\left(\left\lceil\log_{\sqrt{2}}k\right\rceil-\left\lfloor\log_{\sqrt{2}}k\right\rfloor\right)\;.$$
How could you solve this problem? Are there sigma rules or ...
- 59
0
votes
1
answer
28
views
Some trouble with algebra using logarithms and summations
I'm having some embarrassing trouble with algebraic manipulation.
I have the function $$f(y) = y^Tx-\log\sum_{i=1}^ne^{x_i}$$
and for each $i = 1,2,\ldots,n$ $$y_i = {e^{x_1} \over \sum_{i=1}^ne^{...
user192348
5
votes
3
answers
262
views
Using an identity to simplify the sum
So I ran into this problem today. It asks me to use an identity to simplify the sum.
$$\sum_{j=7}^{27}\ln\left(\frac{j+1}{j}\right)$$
I have no idea where to start. I don't know any ...
- 1,185
3
votes
1
answer
5k
views
Relationship between logarithms and harmonic series
This article on the harmonic series says that
$$\sum_{n=1}^k\,\frac{1}{n} \;=\; \ln k + \gamma + \varepsilon_k$$
where
$$\varepsilon_k\sim\frac{1}{2k}$$
and this seems to generalise to
$$\sum_{n=1}...
- 9,018