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

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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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3answers
49 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 ...
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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
48 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} ...
2
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2answers
325 views

Is there a means of expressing this infinite sum?

$$1+22+333+4444+\cdots$$ I honestly do not have any clue on where to even start on this one. What I mean by expressing is with $\Sigma$ symbol and in that form. Thank you very much. After ...
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1answer
110 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, ...
6
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1answer
63 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
39 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
28 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
26 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
60 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
74 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 ...
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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
85 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
124 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?
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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 ...
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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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65 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 ...
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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 ...
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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 ...
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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 ...
10
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4answers
390 views

Is there a function that can be subtracted from the sum of reciprocals of primes to make the series convergent

The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm: $$\gamma = \lim_{n \rightarrow \infty }\left(\sum_{k=1}^n \frac{1}{k} - \ln(n)\right)$$ ...