5
votes
2answers
88 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
66 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 ...
6
votes
2answers
270 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
99 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
12 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
108 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
27 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
37 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
126 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
12 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
121 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
40 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
64 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
86 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 ...
5
votes
3answers
89 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
110 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
100 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
117 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
134 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
41 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
83 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
59 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
140 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
247 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
65 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
60 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.
8
votes
0answers
92 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
457 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
51 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
192 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
209 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
79 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
308 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
96 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
102 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 = ...
3
votes
2answers
159 views

Convergence of integral of log and sum the mean of the logs

How can I show that the following limit converges and $L \in (0, +\infty)$? $\lim\limits_{n \to +\infty}\left( S_n - T_n\right)$, where $S_n = \int\limits_1^n \log x\, dx$, and $T_n = \sum_{k = ...
2
votes
0answers
35 views

Compute $ \operatorname{Li}_{3}\left(\frac{1}{2} \right) $

Where could I find a proof of the identity $$ \operatorname{Li}_{3}\left(\frac{1}{2} \right) = \sum_{n=1}^{\infty}\frac{1}{2^n n^3}= \frac{1}{24} \left( 21\zeta(3)+4\ln^3 (2)-2\pi^2 \ln2\right)$$ ?
1
vote
2answers
250 views

Find the limit of the sequence containing logarithm??

Find $\lim_{n→∞} [log(2+3^n)]/2n$ I have my work till the very last step then i dont know how to continue $\lim_{n→∞} [log(2+3^n)]/2n$ =$\lim_{n→∞} log(3^n)+\lim_{n→∞} log[(2+3^n)/3n]$ ...
1
vote
1answer
825 views

Another two hard integrals

Evaluate : $$\begin{align} & \int_{0}^{\frac{\pi }{2}}{\frac{{{\ln }^{2}}\left( 2\cos x \right)}{{{\ln }^{2}}\left( 2\cos x \right)+{{x}^{2}}}}\text{d}x \\ & \int_{0}^{1}{\frac{\arctan ...
2
votes
1answer
101 views

Some log and exponential integral

I think these should hav some closed form: $$\displaystyle\begin{align*} & \int_{0}^{1}{\frac{\left( 1-x \right)\ln \left( x \right){{\text{e}}^{-x}}}{\pi -x}}\text{d}x \\ & ...
2
votes
5answers
499 views

Convergence of series $\sum_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$?

I have the series $$\sum\limits_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$$ I'm trying to find if the sequence converges and if so, find its sum. I have done the ratio and root test but It ...
2
votes
2answers
81 views

Explanation of this inequality

Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ? I can't see why the inequality is true. I ...
2
votes
3answers
998 views

Taylor series expansion of base 2 logarithms

Sorry for the noob question, but I've been hitting my head against the wall on this for a while. I am looking for a Taylor series expansion of a logarithm other than the natural logarithm $ln(x)$. It ...
1
vote
1answer
77 views

What kind of series / recursion is this?

I'm trying to find the explicit solution / sum of first n elements for the following sequence: d(2) = 2 d(n) = d(n/2) + n*log2(n) Can you help me to find out ...
0
votes
1answer
76 views

Approximating a simple sum

Can somone help me find an assymptotic formula for n, for fixed x , for this sum , perhaps an inequality would be even better, or some bound on the error. $$\sum_{k=1}^n \frac{1}{\log(kx)}$$ I need ...
8
votes
1answer
251 views

Information on the sum $\sum_{n=1}^\infty \frac{\log n}{n!}$

In my personal study of interesting sums, I came up with the following sum that I could not evaluate: $$\sum_{n=1}^\infty \frac{\log n}{n!} = 0.60378\dots$$ I would be very interested to see what ...
1
vote
1answer
115 views

Big O with Log base equivalences and a question about sum of series.

Hi I need help figuring these out: True or False: $\log_2 n$ is $O(\log_3 n)$ I used the definition of Big O in Dasgupta's book: ${f(n)}\over g{(n)}$ $\leq c$ So I used the base transformation rule ...
3
votes
3answers
97 views

Determining and (dis)proving if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges

I am trying to determine if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges using an alternating series test. The test in question requires me ...