0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
3
votes
3answers
49 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
8
votes
1answer
70 views

A closed form for $\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $

This is another 'arctanlog' series: $$ S=\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1) $$ Maybe differentiating with respect to some parameter could be of interest. What ...
11
votes
0answers
136 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
1
vote
1answer
41 views

Series involving a Logarithm

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what ...
1
vote
2answers
63 views

Summation of series involving logarithm: $\sum (n+2)\ln 2^n$

The following question is: Show that $\sum\limits_{r = 1}^n {r(r + 2)} ={n \over 6}(n+1)(2n+7).$ Using this results, or otherwise, find, in terms of $n$, the sum of the series ...
2
votes
4answers
91 views

Is there any expansion for $\log(1+x)$ when $x\gt 1$?

Is there any expansion for $\log(1+x)$ when $x\gt 1$ ?
1
vote
1answer
37 views

Asymptotics of logarithm: $\frac{1}{n}\ln(a+o(1)) = \frac{1}{n}\ln(a)+o(\frac{1}{n})$

I am having problems with the use of the little oh notation my professor is adopting in the solutions to some exercises. As an example I do not understand why $$ \frac{1}{n}\ln(a+o(1)) = ...
1
vote
3answers
68 views

Using Riemann sums to show that $\sum_{i=1}^n 1/i = \log{n} +c+O(1/n)$

I want to show that there exists a constant $c$ such that: $$ \sum_{i=1}^n 1/i = \log{n} +c+O(1/n) $$ I am thinking about Riemann sums. Any hints on that?
0
votes
2answers
39 views

How to use the Comparison Test to investigate the convergence of $\sum (\ln n)/n^\alpha$?

Let $$\sum\limits_{n=1}^\infty \frac{\ln n}{n^\alpha}, \alpha\in\Bbb{R}$$ I need to investigate the convergence of this series. I've read that since the series is positive for all $n$ then it ...
3
votes
1answer
28 views

Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$?

while playing around with my equations, i found that the following has to hold for my universe to be consistent: $$\lim_{N \rightarrow ...
8
votes
1answer
118 views

Feynman's Algorithm for computing a logarithm of a number in [1,2]

I came upon the following quote concerning Feynman (the entire essay this is from can be found here): Consider the problem of finding the logarithm of a fractional number between 1.0 and 2.0 (the ...
5
votes
2answers
109 views

How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

Let $\psi := \Gamma'/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha ...
2
votes
1answer
69 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...
9
votes
2answers
336 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
3
votes
3answers
108 views

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ In the procedure I followed, I came across the following sum: ...
0
votes
0answers
23 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
2
votes
3answers
113 views

A series converging (or not) to $\ln 2$

I have come across the following series, which I suspect converges to $\ln 2$: $$\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}.$$ I could not derive this series from some of the standard ...
0
votes
1answer
29 views

Mathematics - geometric progression question

If $a$, $b$ and $c$ are in geometric progression, then what are $\log_ax$, $\log_bx$ and $\log_cx$ in? What I did: I substituted values for $x, a, b$ and $c$ and tried to solve it further. What I ...
0
votes
0answers
39 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
0
votes
0answers
128 views

A uniformly convergent series

How does one show that the series $$\sum_{k = 1}^\infty \left\{\frac{s}{k} - \log\left(1 + \frac{s}{k}\right)\right\}, \quad s \in \mathbb{C} \setminus \{0, -1, -2, \ldots\}$$ is uniformly convergent? ...
1
vote
1answer
18 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
1
vote
3answers
130 views

Convergence of series minus logarithm

im trying to solve this problem since two, three days.. Is there someone who can help me to solve this problem step by step. I really want to understand & solve this! $$ Show\ \exists \ \beta ...
0
votes
0answers
41 views

Approximation for the logarithm of a summatory

I would like to find an approximation for: $$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$ with $$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + ...
5
votes
2answers
65 views

How to find the limit of $\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$

How to find the following limit: $$\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$$ where $H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n^{th}$ harmonic number and $\gamma$ is the Euler ...
0
votes
1answer
46 views

Prove $\displaystyle n\log\left(1+\frac{1}{n}\right) - \log\left(1+\frac{1}{n+1}\right) < \log\left(1+\frac{1}{n+1}\right)$

I'm trying to prove the above inequality, assuming $n\ge1$. I've been working on this one using log properties and trying to reduce this inequalitiy to simpler ones. Though!, is it even correct? or am ...
1
vote
1answer
96 views

How do computers calculate the log of a value? [duplicate]

I'm not sure if this question belongs on StackOverflow or here (please let me know if the former, and i'll delete this and ask there), but I was wondering how the ...
1
vote
1answer
24 views

Testing the convergence of the series $\sum 1/(k^q (\ln k)^p)$

Determine all values of $p$ and $q$ for which the following series converges: $$\sum_{k=2}^{\infty} \frac{1}{k^q (\ln k)^p}$$ Hints : Consider the three case $q>1$, $q=1$, $q<1$. I ...
5
votes
3answers
93 views

What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger ...
0
votes
1answer
144 views

What is the limit of difference between harmonic series and natural logarithm of n+1?

I'm an undergraduate student in geology and I'm dealing with a project in math. The last question of the project gives me the harmonic series (An = 1 + 1/2 + ... + 1/n) and this natural logarithm L = ...
2
votes
3answers
101 views

Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$

I want to show that for all $a \in \mathbb{R }$ $$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$ So far i've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i ...
6
votes
3answers
143 views

Series comparisons and logarithms?

Prove the convergence of $$ \sum_{n = 1}^{\infty} {\sqrt{\, 2n - 1\,}\,\ln\left(4n + 1\right) \over n\left(n+1\right)} $$ I've been struggling for hours on this. By the textbook we have the limit ...
2
votes
1answer
196 views

Show root test is stronger than ratio test

Let $a_n$ be a sequence of real positive numbers. Show $$\liminf_{n\to  \infty} \frac{a_{n+1}}{a_n} \leq \liminf_{n\to \infty}\,(a_n)^{1/n}  \leq \limsup_{n\to \infty} \, (a_n)^{1/n} \leq ...
1
vote
1answer
46 views

Finding the value of a logarithmic expression involving an infinite GP

Find the value of $(0.16)^{\displaystyle\log_{2.5}(\frac13+\frac1{3^2}+\frac1{3^3}+\cdots)}$. I could solve the series. It gave $$(0.16)^{\log_{2.5}0.5}$$ Unable to proceed from here.
1
vote
1answer
42 views

Maclaurin series of $\ln(2+x^2)$

Find the Maclaurin series of $\ln(2+x^2)$. I know that $\displaystyle\ln(1+x) = \sum_{n=1}^\infty\frac {(-1)^{n-1}} {n} x^n $ So is $\displaystyle\ln(1+x^2) = \sum_{n=1}^\infty \frac ...
2
votes
1answer
84 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
3
votes
1answer
63 views

Simplifying this logarithm series

$$\sum_{i\; =\; 2}^{99}{\frac{1}{\log _{i}\left( 99! \right)}}$$ How would you evaluate (or at least simplify) this logarithm series?
1
vote
3answers
146 views

Does $\sum_{n=2}^\infty (n\ln n)^{-1}$ diverge?

Is $\sum_{n=2}^\infty (n\ln n)^{-1}=\infty$ ? This seems like elementary calculus, but I can't figure this out. Can anyone supply a hint?
5
votes
1answer
260 views

Series involving Logs

I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g. $$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$ I realise that this comes from $x^y ...
3
votes
1answer
66 views

Next number that is a 1 and zeroes

There's probably a much better way of expressing this, but I don't know it, so I guess that's part of the question too. I'm not even sure what to tag it. How do I find the next number greater than a ...
0
votes
1answer
63 views

What is the sum of this series?

The series is as follows: $$\lg n + lg (n - 2) + lg (n - 4) + \ldots + lg (2)$$ Thanks.
9
votes
1answer
118 views

On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
25
votes
1answer
470 views

Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let ...
2
votes
1answer
52 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
5
votes
3answers
193 views

Computing the value of logarithmic series: $Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $

Given a series of the type: $$Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $$ How does one evaluate it? Something I noticed was: $$Q(1,n) = \ln(1) + \ln(2) + \ln(3)+ \cdots+\ln(n) = ...
16
votes
1answer
219 views

Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$

Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
3
votes
1answer
80 views

What is the closed formula for the following summation?

Is there any closed formula for the following summation? $$\sum_{k=2}^n \frac{1}{\log_2(k)}$$
1
vote
3answers
351 views

Exponential and logarithmic series: Find the sum of $2^2 + 3^2/ 2!+4^2/3!+…$ to infinity

Find the sum of the following series: $ 2^2 + 3^2/2! + 4^2/3! + ...$ to infinity The answer is given as $5e$ but I got it as $5e+1$ $T_n = 1/(n-2)! +3/(n-1)! + 1/(n)! $ for $n \ge 2$ and ...
6
votes
3answers
111 views

Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$

Does anybody know how to prove this series? $$\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$$ I arrived at this through Mathematica. I tried writing ...
4
votes
3answers
111 views

How to solve infinitely nested logarithms

I have an iterative process that starts with $$x_1 = \log_{10}(a)$$ Following iterations are as follows: $$x_2 = \log_{10}(a-b\cdot x_1)$$ $$x_3 = \log_{10}(a-b\cdot x_2)$$ $$x_4 = ...