8
votes
1answer
129 views

How prove this sequence $a_{n}=\sqrt{n}+\frac{1}{2}-\frac{1}{8\sqrt{n}}+o\left(\frac{1}{\sqrt{n}}\right)?$

let sequence $\{a_{n}\}$ such $$a_{1}=1,a_{n+1}=1+\dfrac{n}{a_{n}}$$ show that: $$a_{n}=\sqrt{n}+\dfrac{1}{2}-\dfrac{1}{8\sqrt{n}}+o\left(\dfrac{1}{\sqrt{n}}\right)?$$ This result is china student ...
0
votes
0answers
32 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 + ...
0
votes
0answers
31 views

Is there a way to expand Re(Li(a^z)) in series?

I'm searching a way to expand $ f(z) = Re(Li(a^z)), a \in R, z \in C $ in series. The computer-friendly, quickly convergent series is a huge plus. For being 'computer-friendly' I mean a relatively ...
1
vote
2answers
48 views

Why sum of sigmoids is a good approximation of softplus function?

According to this paper: Rectified Linear Units Improve Restricted Boltzmann Machines, $\sum_{i=1}^N \sigma(x-i+\tfrac{1}{2}) \approx \log(1+e^x)$ (equation 7) where $\sigma(z) = \frac{1}{1+e^{-x}}$ ...
0
votes
1answer
23 views

How to find sequence from set of values

I have set of probability values arranged in ascending order, p1<p2<p3<...<pM. Now I want to assign set of numbers in the same manner in which probabilities are increasing. It means I ...
1
vote
1answer
84 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
6
votes
4answers
489 views

Evaluating the precision in the calculation of $\mathrm{e}$

I'm calculating $\mathrm{e}$ using a computer like this: $$ \mathrm{e} \approx \sum\limits_{i=0}^n {1\over i!} $$ I'm storing it as a rational number. I was wondering, if I write down my rational ...
2
votes
1answer
45 views

Approximating $\tanh(B\sqrt{A} )$ for small $A$ and arbitrary $B$ by correcting for asymptotic behaviour of $\tanh$

I would like to approximate a function containing terms of the form $\tanh( B\sqrt{A})$ for small $A$. I have tried doing a Taylor series, but I consistently find that it is not only $A$ that has to ...
1
vote
1answer
48 views

How to find a sequence by its limit?

Is there any way to construct non-trivial sequence by its limit? Something like $\begin{cases} a_1=2 & \\ a_{n+1}=\dfrac1{2}\left(a_n+\dfrac2{a_n}\right) \end{cases}$for $\sqrt2$. I'm especially ...
3
votes
1answer
86 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) ...
3
votes
2answers
74 views

Approximating $e^{-x}$

So we all know that the Taylor series for $e^x$ is $1 + x + \frac{x^2}{2} + \frac{x^3}{6}+\ldots$ Similarly for $e^{-x}$ it comes comes out to be $1 - x + \frac{x^2}{2} - \frac{x^3}{6}+\ldots$ Now for ...
1
vote
1answer
26 views

Showing that $\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0$ but the matching series does not converge

I want to show that: $$\lim\limits _{n\to\infty}{n \choose \left\lceil \frac{n}{2}\right\rceil }\cdot2^{-n}=0 $$ And also $${\displaystyle \sum_{n=1}^{\infty}{n \choose \left\lceil ...
3
votes
1answer
113 views

$n \approxeq k + 2^{2^k}(k+1)$. How can one get the value of $k(n)$ from this equation?

I am trying to find approximation for this sum. Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$ Doing following way. Let $a_k(n) = \frac{n\choose k}{2^{2^k}}$. I tried to ...
3
votes
1answer
185 views

Asymptotic approximation of sum $\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$

I need to find the asymptotic approximation of this sum $$\sum_{k=0}^{n}\frac{{n\choose k}}{2^{2^k}}$$ Can you please share a link to theory or hint how it can be solved? Here is my attempt $n ...
1
vote
2answers
84 views

How to approximate this series?

How to approximate this series, non-numerically? $ S_n = \sum_{n=1}^{50} \sqrt{n}$
2
votes
1answer
44 views

Why does this pattern fail (sometimes) for the continued fraction convergents of $\sqrt{2}$?

This is connected to my post on the continued fraction convergents of pi. Motivated by Calvin Lin's comment whether a similar pattern exists for other constants, I checked $\sqrt{2}$. Its convergents ...
9
votes
1answer
322 views

A strange “pattern” in the continued fraction convergents of pi?

From the simple continued fraction of $\pi$, one gets the convergents, $$p_n = \frac{3}{1}, \frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102}, \frac{104348}{33215}, ...
9
votes
1answer
125 views

Is $e = \sum_n 1/n!$ the most efficient sequence of denominators for rational series for $e$?

The classical series $e = \lim_{n \to \infty} X_n$ where $X_n = \sum_{k=0}^n 1/k!$ is incredibly efficient. But is it known to be the most efficient series in terms of denominators for using fractions ...
0
votes
3answers
81 views

Sum of sequence precision

I came up with this answer in stackoverflow. It states a question: ...
2
votes
1answer
59 views

What information is needed to solve or approximate this simple equation?

Suppose I have some vector $x = a + b = (a_0 + b_0, \dots, a_{n-1} + b_{n-1})$ of length $N$. Now, if only $x$ is known, what is the minimum amount of information that is needed to either solve or ...
1
vote
0answers
88 views

Turn ugly series into a nice approximation

I am currently struggeling with a series(actually two). The problem is, that I can do nothing with them, since this expression is so ugly. I would love to hear about any kind of approximations that ...
1
vote
1answer
86 views

Harlan J. Brothers's approximation to $ e $ ad infinitum?

Consider the series generated by Harlan J. Brothers's method for the number $e$ http://en.wikipedia.org/wiki/List_of_representations_of_e Can they be improved again and again or is there a limit so ...
0
votes
1answer
79 views

Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?

I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a ...
2
votes
3answers
52 views

Series evaluated to $m$ terms, approximating the error

Given a series $\displaystyle\sum_{n=0}^\infty a_n$, how can we bound the error (which I shall denote with $R_n$) when we evaluate it to $m$ terms? $$\sum_{n=0}^\infty a_n \approx \sum_{n=0}^m a_n$$ ...
1
vote
0answers
77 views

Expansion in powers

Let $n=2k, k \in Z_+$. Let $$P_k\left(\frac{t}{\sqrt n}\right)=n!\sum_{\begin{smallmatrix} n_1+\ldots+n_k=n \\ ...
1
vote
0answers
53 views

alternating series estimation with integral?

We know that there are some approximation like Abel's identity. If $\lambda_n$ is increasing and $$ C(x)=\sum_{\lambda_n\le x}c_n,\qquad(c_n\in\mathbb{C}) $$ Then if $X\ge\lambda_1$ and $\phi(x)$ has ...
1
vote
0answers
67 views

Intuition for approximating Ei(x)

I'm working with a function that is defined to be the sum of a series, and I'd like to either find its values, or approximate them analytically: $F(x) = \frac{1}{w} ...
5
votes
1answer
193 views

Approximating a weird sum

How can I approximate the sum$$\sum_{k=1}^n \left(\frac{2k}{n} \left\lceil \frac{n}{k} \right\rceil \left\{ \frac{n}{k} \right\}-1\right)$$ where $\{x\}$ is the fractional part function, and $\lceil ...
2
votes
1answer
48 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
8
votes
1answer
170 views

Is this sequence convergent?

I heard this question from a professor a couple years ago. I still think about it... Does the sequence $(a_n)_{n\in \mathbb N}$ with $$a_n=\sqrt[n]{|\sin(n)|}$$ converges ( to $1$ ) ? I believe ...
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 ...
0
votes
0answers
82 views

Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
1
vote
1answer
50 views

How to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?

Is there some standard way to approximate a complex linear homogenous recurrence with constant coefficients with a simple one? For example, I might want to approximate $$ ...
10
votes
2answers
341 views

Summation of divergent series of Euler: $0!-1!+2!-3!+\cdots$

Consider the series $$\sum\limits_{k=0}^\infty (-1)^kk!x^k\in\mathbb{R}[[x]]$$ Let $s_n(x)$ be partial sum. And let $\omega_{k,n}=(k!^2(n-k)!)^{-1}$. My question is: Prove that ...
1
vote
1answer
326 views

Using binomial theorem find general formula for the coefficients

Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion: $$ ...
1
vote
1answer
353 views

Approximating the logarithm of sum

I would like to approximate $$ \ln(\sum_{k=0}^n(n-2k)^p) $$ Here $p\geq 2$
0
votes
0answers
110 views

Calculation of sum

I am wondering if it is possible to calculate or approximate the following sum $$ \sum_{k=0}^l\frac{(l-2k)^p(2l+k(k-1))l^{k-1}}{(k+3)(k+2)} $$here $p \geq 2$. Thank you.
1
vote
0answers
283 views

Calculation of a 'double' sum

Let $n \in N$ and $q\geq 2$. I am trying to calculate the following sum: $$ \sum_{i=0}^{\sqrt n/2}\sum_{j= i \sqrt n }^{(i+1)\sqrt n}\frac{(-1)^q2^q(\frac{n}{2}-j)^q}{(n-j)!j!} $$ Any help will be ...
2
votes
1answer
174 views

Best and most efficient way to numerically compute $e$?

There are many well-known methods for efficiently numerically computing $\pi$, such as Chudnovsky's Method or perhaps Gauss-Legendre's algorithm. I was wondering what the best method for computing $e$ ...
1
vote
1answer
236 views

Newton-Raphson's Method to find $\sqrt{2012}$

I am asked to find $\sqrt{2012}$ using Newton-Raphson's Method with the following recursive method $$x_{n+1} = \frac{1}{2} (x_n + \frac{a}{x_n}) $$ I notied that give same answers as using ...
3
votes
2answers
458 views

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I ...
8
votes
4answers
208 views

How could we manually approximate $\sum_{i=1}^{50} i! = 3.1035 \times 10^{64}$?

How could we manually approximate $$\sum_{i=1}^{50} i!$$ to the value $ 3.1035 \times 10^{64}$? I faced this question in my aptitude test,there were four option given,I couldn't solve it during ...
27
votes
1answer
610 views

A series problem by Knuth

I came across the following problem, known as Knuth's Series which originally was an American Mathematical Monthly problem. Prove that $$\sum_{n=1}^\infty ...
1
vote
2answers
145 views

Find fast exact value for numbers in the form $\sum_{k=min}^{Max}\frac{1}{k}$

I know I could start multiplying by all denominators and try to get the exact value that way but is there some smarter way or shortcut? Let's take simple example: $\displaystyle ...
5
votes
3answers
428 views

$\lim_{n\to\infty} f(2^n)$ for some very slowly increasing function $f(n)$

I should be able to answer this myself, but feel insecure anyway. I want to know, whether a function f(n) is bounded if n goes to infinity (and if it's bounded, the limit). Heuristically it appears ...
36
votes
4answers
3k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
2
votes
3answers
187 views

Determine speed of the object at the current time by the non-uniform time sample

Here is a time sample: $Q = \{(t_i, x_i) | 0 \leq x_i \leq x_{i+1}, 1 \leq i \leq n\}$ and rules: (1) $T_1 \leq t_{i+1} - t_i < T_2$ where $T_1, T_2 > 0$ (2) $x_i$ comes with error: $x_i = ...