For questions about recurrence relations, convergence tests, and identifying sequences

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3 views

Series expansion of reciprocal function of Generalized Exponential Integral

Generalized Exponential Integral of order p has a series expansion http://dlmf.nist.gov/8.19.10 Is there a series expansion of the reciprocal function ?
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0answers
13 views

Regularization of a (divergent) cosine series

What would be a suitable regularized value for the following divergent series: $$ S(y) = \sum_{k=1}^{\infty} \cos(k y) \quad y \in R\\ $$ By way of added context, this series arises from a formal ...
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1answer
25 views

Uniform convergency of a two sequence of functions

For, $n\ge 1$ let, $g_n(x)=\sin^2(x+1/n)$ , $x\in [0,\infty)$ and $f_n(x)=\int_0^xg_n(t)\,dt.$ Then, which is(/are) correct ? (A) $\{f_n(x)\} $ converges pointwise to a function $f$ on $[0,\infty)$ ...
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4answers
63 views

$0,1,0,1,0,1$… has only $2$ limit points

Prove that the sequence $0,1,0,1,0,1$... has only $2$ limit points : $0$ and $1$. To be frank, I know the solution to the above particular problem. What I am interested is in knowing a general ...
2
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2answers
26 views

Finding error in reasoning about geometric series sum

A supposedly fast method to find the sum of a geometric series is the following one. Let $$S = \sum_{n = 0}^{+\infty} q^n$$ then $$S = 1 + q\left[\sum_{n = 0}^{+\infty} q^n\right] = 1 + qS.$$ Hence ...
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1answer
75 views

Proving a convergence relationship between two sequences

Let $a_{n}$ a sequence of real numbers. Let $\sigma_n= \frac{a_1+a_2+...+a_n}{n}$. Suppose that $\lim_{n\to \infty} \sigma_n=A.$ Prove that $$\lim_{n \to \infty}\frac{1}{\log n} ...
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2answers
515 views

Expressing the infinite sum 1 + 22 + 333 + 4444 + …

I would like to express $$1+22+333+4444+\cdots$$ using $\Sigma$ notation, and have no clue where to start. After $999999999$, comes 10 $0$s, then 11 $1$s.
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1answer
115 views

Find all values such that the series converge:

Find all the values of $p\in\mathbb R$ such that the following series converge: $$\sum_{k=2}^\infty (\log k)^{p\log k}$$ I would like hints only. I've tried using the exponential function ...
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1answer
37 views

What is value of k in next hermonic serie? [on hold]

I want to know the value of k such that $\sum_{n=1}^{k}\frac{1}{n} \geq 8000$. Thanks.
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1answer
38 views

Prove this sequence is bounded

This is not any exercise on itself, but I was reading a proof in which a sequence similar to this appeared: $$a_n=n\lambda^{n-1}, \quad|\lambda|<1$$ In essence. Then I came across the assertion, ...
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1answer
64 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
3
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1answer
47 views

Strategies For Summing Harmonic Numbers

Lately, I have found several interesting problems involving Harmonic numbers such as \begin{equation*}\sum_{n=1}^{\infty}\frac{H_n^{(2)}}{n^2}=\frac{7\pi^4}{360}\end{equation*} I am not familiar with ...
3
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1answer
29 views

Identities For Generalized Harmonic Number

I have been searching for identities involving generalized harmonic numbers \begin{equation*}H_n^{(p)}=\sum_{k=1}^{n}\frac{1}{k^p}\end{equation*} I found several identities in terms of $H_n^{(1)}$, ...
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1answer
29 views

How do I find if a series of integers is (somehow) regular

[DISCLAMIER]: please be patient and low profile: no math background here :) I have a database of transactions, i.e. a list of purchases made by different customers. What I have to figure out is ...
3
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0answers
64 views

Computing $\sum _{k=1}^{\infty } \frac{\Gamma \left(\frac{k}{2}+1\right)}{k^2 \Gamma \left(\frac{k}{2}+\frac{3}{2}\right)}$ in closed form

What tools other than beta function you might like to use here? $$\sum _{k=1}^{\infty } \frac{\displaystyle \Gamma \left(\frac{k}{2}+1\right)}{\displaystyle k^2 \Gamma ...
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4answers
63 views

Find the limit $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$ [on hold]

I need help finding the following limit. $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$$
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1answer
42 views

I need to find an nth term for a fairly random sequence [on hold]

I need an nth term of the sequence 6,2,0,3,1,8,10,5,6,6,4,5,4,6 there is no method for finding the next numbers, all i need is an nth term which will work. thanks :D
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1answer
42 views

Recurrence relation involving infinite sequences.

How in general one would solve an infinite series recurrence relations? For instance, I am interested to solve the following: \begin{equation} \sum_{n=0}^{\infty} (-1)^{n} F(n)\{1-(\alpha n ...
6
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1answer
95 views

Does $ \sum_{(m,n) \neq (0,0)} \frac{(-1)^{m+n}}{m^2 + n^2} $ have an exact value?

I am looking for an $\mathbb{Z}[i]$ analogue of the alternating harmonic series: $L(1,\chi)=\sum_{n=0}^\infty (-1)^n \frac{1}{n} = \frac{\pi}{4}$. If we try adding the reciprocals of the Gaussian ...
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0answers
12 views

How to optimize the repartition of samples in weighted channels?

This is more like an applied mathematics question, so my apologies if I am at the wrong place. Let S(n) be an infinite sequence of real numbers strictly growing from 0 to 1 (asymptotically). Let P be ...
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4answers
56 views

Show $ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$

I am tasked with proving the following limit: $$ \lim_{n\rightarrow \infty} 2^{-1/\sqrt{n}}=1$$ using the definition of the limit. I think I have done so correctly. I was hoping to have someone ...
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2answers
76 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
2
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2answers
170 views

Help with Convergence of a series with sin and log

I tried to figured it out if the follwing series converges or not $$\sum_{n=1}^{\infty} \frac{\sin(\frac{1}{n})}{\ln^2n}\ (-1)^{n}$$ I tried to show that $\sin(\frac{1}{n})$ is a monotonic but I'm ...
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1answer
30 views

Discuss the convergence of $\sum a^n/(x^n+a^n)$?

I fail to understand how any of the tests I know will work on this. I tried the D'Alembert's Ratio test but I don't understand how it will limit to something.
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5answers
86 views

Proving that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$

I was reading sequences in Terence Tao Analysis book and I came across the question: Prove that $\forall x>0$, $\lim\limits_{n\to\infty}x^{1/n}=1$ In the hint it says that ... you may ...
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1answer
20 views

Non-Cauchy products for series?

Let $\sum a_n$ and $\sum b_n$ be two absolutely convergent series. My question is Is it possible to take the product of the two series $(\sum a_n)(\sum b_n)$ and get a result different from the ...
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3answers
26 views

Should I use the comparison test for the following series?

given the following series $\sum_{k=0}^\infty \frac{\sin(2k)}{1+2^k}$ I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would ...
3
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2answers
125 views

Convergent by Ratio test?

I am lost with this problem: $$\sum_{n=1}^\infty \frac{n^n}{2^n n!}.$$ I am suppose to find if it is convergent or divergent. I have the correct set up. After cancelling everything I am left with ...
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3answers
78 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
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4answers
209 views

Is the following Alternating Series Absolutely Convergent?

$$\sum_{n=1}^\infty\frac{(-1)^n}{2n+1}$$ I think it is Absolutely Convergent because it converges by direct comparison to Harmonic series? Am I right or wrong?
2
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0answers
38 views

How could one invert this sum of Stirling numbers?

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
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2answers
28 views

If the i-th, j-th, and k-th terms in an AP are in a GP with ratio r, find $r$ in terms of $i, j$, and $k$

If the i-th, j-th, and k-th terms in an arithmetic progression are in a geometric progression with ratio r, find r in terms of i, j, and k. This is my result: (1) if $ik \ne j^2$ then ...
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0answers
58 views

How $\zeta(-1)$ is defined? [duplicate]

I know some proof that $\zeta(-1)$ equals to $-\frac{1}{12}$. here it is: Let $S_1 = (1)+(-1)+(1)+(-1)+...$ Then we have that $2S_1=1$ because if we shift the second $S_1$ for 1 to right we have ...
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0answers
48 views

Fibonacci series generated with division $1/999999999999999999999998999999999999999999999999$ [duplicate]

I tested the suggestion of finding the Fibonacci series by division, which sounded very surprising to me. I therefore used a simple sympy script to test it and found that it works as advertised. ...
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1answer
69 views

Using the ratio test when the limit of ratio is infinity

If the limit in ratio test is infinity. Does the sequence converge? I suspect not as it is infinity and not some finite value but I'm not sure. Any help?
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4answers
131 views

How to prove that series $\frac{1}{n+1}$, as $n\to \infty$ is zero. [on hold]

Can somebody explain how to prove that series $\frac{1}{n+1}$, as $n \to \infty$? I mean infinite series, not sequence, and I want to understand how to define the partial sum when n goes to infinity. ...
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6answers
122 views

how to prove that $\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$ [duplicate]

I need to prove $$\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$$ For now, I only know the e definition: $\lim\limits_{n\to\infty} \left ( 1+\frac{1}{n} \right)^n = e $, and I ...
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2answers
33 views

Prove that $\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$

I'm trying to prove by induction that $$\tan \left ( \sum_{k=1}^{n} \theta_k \right ) \geq \sum_{k=1}^{n} \tan (\theta_k)$$ provided that $$\sum_{k=1}^{n} \theta_k < \frac{\pi}{2}$$ So in ...
3
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3answers
49 views

Expanding $\frac{2x^2}{1+x^3}$ to series

So I was doing some series expansion problems and stumbled upon this one ( the problem is from Pauls Online Notes ) $$f(x) = \frac{2x^2}{1+x^3}$$ The actual solution to this problem uses a ...
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2answers
83 views

Two divergent series such that their product is convergent

I faced a series question it goes something like give an example of 2 divergent series such that when the 2 series are multiplied to each other, the new series becomes convergent, although it looks ...
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0answers
69 views

Show a function defined by summation is increasing, another is decreasing

Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows. $f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ , ...
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1answer
19 views

Sequence uniform convergence but the derivatives are not.

Give a sequence $(f_n)_{n\in \mathbb{N}}$ of differentiable functions which uniformly converge to $0$, but for which the seqeunce $(f_n')_{n\in \mathbb{N}}$ of the derivatives isn't even pointwise ...
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5answers
131 views

A limit problem: $\lim\limits_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$

I need help in solving the limit below: $$\lim_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$$ What I've done is to simplify ...
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2answers
70 views

Express the number $4$ and $5$ and $6$ and $7$ and $8$ [on hold]

Express the number $4$ and $5$ and $6$ and $7$ and $8$ with trigonometric identities or series or equations. example: Express the number $1$, $$\cos^2 x + \sin^2 x=1$$ Express the number $2$, ...
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1answer
27 views

Infinite sub-sequences that make up a sequence

A sequence $\{a_n\}$ can be broken into sub-sequences, $\{a_n\}^1_{k_1}, \{a_n\}^2_{k_2}, \dots,\{a_n\}^m_{k_m}$, if every element in $\{a_n\}$ belongs to at least one of the sub-sequences. I had to ...
0
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1answer
43 views

How can I raise a Taylor Series to a power?

I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the ...
1
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1answer
91 views

Definition of exponential function -

A lot of textbooks offer a definition of the exponential function such as this: $$\exp(x):=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$$ a) Show that the given definition for $\exp$ is correct, ...
1
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1answer
28 views

Integral approximation for alternating series

I can approximate the sum of $\frac 1 {n^2}$ using its integral. But what about $(-1)^n\frac 1 {n^2}$? Is it possible to approximate this using integrals? I want to know if there are other ways than ...
1
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4answers
38 views

Closed set in $l_{2}$

I need to show that the set $A=\left\{ x \in l_{2} : |x_{n}| \leq \frac{1}{n}, n=1, 2, ...\right\}$ is a closed subset of $l_{2}$ I'm assuming the best way to show this is to have a sequence in A ...
3
votes
1answer
36 views

Analytic continuation of a certain Dirichlet series

Is there an elementary way to analytically continue $$f(s)=\sum_{n=1}^\infty \frac{(-1)^n}{(2n+1)^s}$$ to the entire complex plane? It is not hard to see (by grouping terms in pairs and using the ...