For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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0
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3answers
59 views

Convergence of arithmetic mean of first $n$ numbers of a convergent sequence

Given any convergent series $(a_n)_{n \in \mathbb{N}}$,consider a new sequence $(b_n)_{n \in \mathbb{N}}$, defined as $$b_n = \frac{1}{n}(a_1 + a_2 + \dots+a_n)=\frac{1}{n}\sum_{k=1}^na_k $$ for every ...
1
vote
1answer
36 views

Lim of $a_{n} = (1-\frac{1}{2^2})(1-\frac{1}{3^2})…(1-\frac{1}{n^2})$

$a_{n} = (1-\frac{1}{2^2})(1-\frac{1}{3^2})...(1-\frac{1}{n^2})$ I first find that $a_{n}$ is descending ...
22
votes
1answer
311 views

Odd digits of $2^n$

Let $u_{b}(n)$ be equal to to number of odd digits of $n$ in base $b$. For example: In base $10$, $u_{10}(15074) = 3$ In base $13$, $u_{13}(15610) = u_{13}([7, 1, 4, 10]_{13}) = 2$ What is the ...
0
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1answer
34 views

Limit of a sequence

I've been working on a problem related to convergence of sequences. Specifically, I need to prove that if $a_1, a_2, a_3,... $ is a sequence of real numbers with $\sum^{\infty}_{j=1} |a_j|< \infty$ ...
1
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1answer
67 views

Show monotonicity of an Alternating Series

Let $a_i$, $i= 1,2,..,N$, be a collection of positive numbers, such that $\sum_{i=1}^N \frac{(-1)^i}{a_i} > 0 $ The question asks to show whether the following is true: $$ \sum_{i=1}^N ...
1
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1answer
79 views

Show $\sum_{k=1}^\infty\left(\exp\left(\frac{(-1)^k}{\sqrt{k}}\right) - 1\right)$ diverges

I have a sum that is divergent, but I can't prove it. Can somebody give me a hint how to get this? It should be fairly elementary. The sum is $$\sum_{k = 1}^{\infty} ...
0
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1answer
33 views

Increment of a Positive Series

Let $a_i$, $i= 1,...,N$, be a collection of real numbers, $a_i \neq 0$, such that $\sum_i^N \frac{1}{a_i} > 0 $ The question asks to show whether the following is true: a) $$ \sum_i^N ...
1
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4answers
67 views

Is the interval $[-\frac{1}{n}, \frac{1}{n}]$ equal to $0$ as $n$ goes to $\infty$

Sorry if this is a dumb question, but does the interval $[-\frac{1}{n}, \frac{1}{n}]$ become $0$ as $n$ goes to $\infty$ or does it not quite get there... In other words, does $[-\frac{1}{n}, ...
0
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1answer
49 views

Prove the Limit Superior of a sequence is a real number

($a_n$) is a real sequence bounded from above. Let $A :=$ {$s \in \Bbb R:$ $s$ is the limit of a convergent subsequence of ($a_n$) }. Suppose the supremum of A is a real number, prove by ...
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3answers
50 views

How to find sum of equation from 1 to N

I understand that the sum of n from $1$ to $n$ is $\frac{n(n+1)}{2}$. I'm trying to figure out the sum from $1$ to $n$ of the following expression $\frac{L(n-1)}{R}$ where $L$ and $R$ are unknown ...
1
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1answer
49 views

What use does a Cauchy principal value and divergent summation have?

Through some questionable methods, there lies an answer to the following integral. $$\int_{-a}^a\frac{dx}x=0$$ You may question its soundness at first glance since: ...
2
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2answers
67 views

True of false: The sum of this infinite series. [duplicate]

I'm fairly sure it is false, but I'm not quite sure about which test I should use to prove it. $$\sum_{n=2}^\infty \ln\left(\frac{n-1}{n}\right) = -1 $$ I think using the integral test should work, ...
2
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1answer
57 views

Given arbitrary $c$, which $n$ satisfies $\sum_{k=1}^{n-1}\frac1k<c≤\sum_{k=1}^{n}\frac1k$?

Everyone knows that the harmonic series $$\sum_{k=1}^\infty\frac1k$$ is a monotonically-increasing divergent series. This should imply, I think, that for all $c>1$ there exists some natural $N$ ...
1
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2answers
47 views

Convergence of product of (weakly) converging sequences in $L^{p}$

Preparing for an exam, I was wondering about general statements about the convergence of products. 1) Let $p, q \in ]1, \infty[$ such that $\frac{1}{p}+\frac{1}{q} = 1$ and $a_n \rightharpoonup a$ ...
0
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4answers
148 views

How many numbers are there of 2n digits that the sum of the digits in the first half equals the sum of the digits in the second half

The question is how many number of a given number of digits 2n where the sum of the first half of the digits equals the sum of the digits in the second half. So this is for a programming problem and ...
7
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3answers
170 views

Variety of proofs of a simple proposition in arithmetic

I have a couple of simple proofs of a simple proposition and I'm curious to see the variety of different approaches others would take to prove the same thing. Definition: The dilation of a sequence ...
0
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2answers
29 views

Approximating a sum of reciprocals

What is a good approximation for the function: $$S_{N,k} = \sum_{i=N}^\infty {\frac{1}{i^k}}$$ when $k$ is a given constant (2, 3 or 4) and $N$ is large? $S_{N,k}$ is a decreasing function of $N$; ...
0
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3answers
75 views

Find the limit of the following sequence. (Real Analysis) [duplicate]

Find the limit of the following sequence. $$a_n = \frac{\sqrt[n]{(n+1)(n+2)...(2n)}}{n}$$ I tried couple of methods: Stolz, Squeeze, D'Alambert. But I can not seem to make a conclusion on the limit. ...
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0answers
25 views

The merging lemma for series

(Merging lemma - Maybe it's not called like this in English, but it's how we call it at our university) We'll say that than a family of sequences $(b_{1n}), (b_{2n}), ..., (b_{kn})$ spans the ...
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2answers
56 views

Proof of convergence

I have the following problem I want to solve with induction method. Would be great if someone helped me with it. I have $a_0=0$, $a_1=$$1\over 2$ and $|a_{n+1}-a_n|\le|a_n-a_{n-1}|^2$ I need to show ...
0
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0answers
36 views

Proof of a series law

I'm stuck on the following exercise: "Let $\sum_{n=m}^\infty a_n$ be a series of real numbers, and let $k\geq 0$ be an integer. If one of the two series $\sum_{n=m}^\infty a_n$ and ...
1
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0answers
53 views

Help me find the mistake in my solution of the limit

Let $x_1=1, x_2=2,$ $$x_n=x_{n-1}+x_{n-2}, (n>2)$$ The task is to find: $$\lim \limits_{x\to\infty}\frac{x_{n+1}}{x_n}!$$ My attempt at solution: We write the recursive formula as: ...
0
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0answers
40 views

Limit of a sequence $a_n = \sqrt[n]{n!}/n$ [duplicate]

Find the limit of the sequence $$a_n = \frac{\sqrt[n]{n!}}{n}$$ I can figure out the limit of the sequence by letting $n=1,2,3,\dots$ but what would be the more conceptual approach to finding the ...
1
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3answers
90 views

Prove convergence of $\sum _{n=1}^{\infty }\sin(1/n)/n$ [closed]

Prove convergence of $$\sum _{n=1}^{\infty }\left(\frac{1}{n}\cdot \sin\left(\frac{1}{n}\right)\right)$$. bounded by 2. I have tried a lot, without any success.
1
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1answer
37 views

Explanation of fibonacci like sequence limit solution

So, the task is to find $\lim \limits_{x\to\infty}x_n$, where: $$x_n=\frac{x_{n-1}+x_{n-2}}{2}, n\geq3,x_1=a,x_2=b$$ What my teacher did is: $$2\lambda^2-\lambda-1=0$$ $$(2\lambda+1)(\lambda-1)=0 ...
0
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2answers
24 views

What is the limit of this sequence? Including a cosine

Let the sequence $(a_n)$ be defined by $$ a_n = \frac{ n + \cos(n)}{\sqrt{n^2 + 1}}. $$ I need to find the limit of this. But since the cosine term will oscillate infinitely many times between $-1$ ...
1
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3answers
47 views

Obtaining the value of a power series similar to sine

I apologies for the vague title and the very specific question. I would like to know what $$K=4\left[ \frac{1}{1\cdot2!}-\frac{1}{3\cdot4!}+\frac{1}{5\cdot6!}-\cdots \right]$$ evaluates to. This is ...
1
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0answers
17 views

Upper bound of $\left|\sum_{n,k} c_{n,k}\, \alpha_k\right|^2$.

We have a double finite series, such that: $$\sum_{n,k} |c_{n,k}|^2=1$$ What can we conclude about upper bound of $\left|\sum_{n,k} c_{n,k}\, \alpha_k\right|^2$ for $|\alpha_k|<M$? I have tried to ...
7
votes
1answer
202 views

Evaluate $\displaystyle \sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x.$

I have some trouble in evaluating this series $$\sum_{n=1}^{\infty }\int_{0}^{\frac{1}{n}}\frac{\sqrt{x}}{1+x^{2}}\mathrm{d}x$$ I tried to calculate the integral first, but after that I found the ...
2
votes
3answers
116 views

Calculate the limit $\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+…+\frac{1}{\sqrt{n^2+n}}\right)$

I have to calculate the following limit: $$\lim_{n\to\infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+...+\frac{1}{\sqrt{n^2+n}}\right)$$ I believe the limit equals $1$, and I think I can ...
1
vote
1answer
33 views

How to prove this limit of Airy Function.

I have no idea how to prove this limit $$\lim_{x\rightarrow \infty }\exp\left ( \frac{2x^{3/2}}{3} \right )\sqrt[4]{x}\mathrm{Ai}\left ( x \right )=\frac{1}{2\sqrt{\pi }}$$ where ...
0
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0answers
17 views

Solving $(\prod^t_{i=1} N_i m_i)!$

I would like to know how to solve or simplify the factorial $(\prod^t_{i=1} N_i m_i)!$. Here, $i, N_i, m_i$ are positive integers. My effort: $$(\prod^t_{i=1} N_i m_i)!$$ $$\implies (\prod^t_{i=1} ...
0
votes
1answer
57 views

Sequence of Functions - Pointwise and Uniform Convergence

I'm learning about sequences of functions and need some help with this problem: Show that the sequence of function $f_n(x)$ where $$f_n(x) = \begin{cases} \frac{x}{n}, & \text{if $n$ is ...
0
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1answer
35 views

An intuitive explanation for the negatives of divergent summations?

I am looking for an intuitive explanation for why divergent summations (that are always increasing) have finite values assigned as negative. An example that is beyond "because the math says so" kind ...
1
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0answers
50 views

Use summation by parts to show with certainty that $\sum_{k=1}^\infty (\sin k)/k>0$.

Summation by parts: Let $\{a_k\}$ and $\{b_k\}$ be arbitrary sequences of real numbers, and let $s_n=\sum_{k=0}^n a_k$. Then for $0\le m< n$, ...
0
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2answers
30 views

Convergence of a serie by comparison criteria

I have to look for the convergence of the following series: $$ \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n\sqrt{n+1}} $$ I know I have to make use of the comparison criteria but I do not know how to follow ...
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0answers
20 views

A hypothesis concerning whether or not a divergent summations can be assigned a value.

I hypothesize that the following category of divergent summations cannot be assigned finite values. $$\lim_{w\to\infty}\sum_{n=0}^wa_n=\pm\infty$$ $$\lim_{n\to\infty}a_n-a_{n+1}=0$$ If the ...
3
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1answer
91 views

Is $\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+…$ irrational?

Is there known way to determine whether the infinite sum below is rational or not? $$\sum_{p}\frac{1}{p!}=\frac{1}{2!}+\frac{1}{3!}+\frac{1}{5!}+...$$
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1answer
43 views

Uniform convergence for $ x \in [2, +\infty) $

Could you give me a hint on how to prove that $$ \sum_{n=1}^{\infty} \frac{\log(1+nx)}{nx^n} $$ converges uniformly for $ x\in[2,+\infty) $?
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1answer
38 views

Proving convergence

I want to prove that the following sequence is convergent: $$a_{n+1}=\frac{1}{4(1-a_n)}$$ And $a_0=0$. I should show that the sequence is increasing and bounded. I could not find a way to go about ...
2
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1answer
35 views

Solving $\lim_{n\to\infty}\sum_{k=1}^{n}\frac{2015^{k}}{\sum_{i=0}^{k-1}2015^i\sum_{l=0}^k2015^l}=$

$$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{2015^{k}}{\sum_{i=0}^{k-1}2015^i\sum_{l=0}^k2015^l}=$$ $$\lim_{n\to\infty}\sum_{k=1}^{n}\frac{2015^k}{\frac{1-2015^k}{1-2015}\cdot\frac{1-2015^{k+1}}{1-2015}}=$$ ...
2
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1answer
40 views

Twist on the integral test

I have a question regarding a slight modification of the integral test. Suppose that $a_k = f(k)$ for some continuous function $f:[1,\infty) \rightarrow [0,\infty)$, which satisfies $f(k)\rightarrow ...
0
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1answer
58 views

Pointwise and Uniform converge of $f_n(x) = \frac{nx^2}{1 + nx}, x \in [0, 1]$

I'm learning about sequences of functions and need some help with this problem: Investigate pointwise and uniform convergence of the sequence of functions $$f_n(x) = \frac{nx^2}{1 + nx}, x \in ...
0
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2answers
47 views

Sequence defined by max$\left\{a_n, b_n \right\}$. Proving convergence

Assume $(a_n)$ and $(b_n)$ are two real sequences, and define $$ c_n = \text{max}\left\{a_n, b_n\right\} $$ for $n \in \mathbb{N}$. Suppose $(a_n)$ and $(b_n)$ are two convergent sequences. Prove then ...
0
votes
2answers
30 views

Proving a sequence is bounded from above?

Define the real sequence $(a_n)$ recursively as $a_0 = 1$ and $$ a_{n+1} = 3 - \frac{1}{a_n}. $$ I already proved by induction that this sequence is increasing, i.e. $a_{n+1} \geq a_n$ for all $n \in ...
0
votes
1answer
30 views

nth term test for divergence: Why doesn't it apply to this infinite series?

$$a_n=(-1)^{n+1}$$ $$S_n={1-1+1-1+...(-1)^{n+1}}$$ If $S_n$ sums $S_n$ in the following order:$a_1+(a_2+a_1)+(a_3+a_2)+(a_4+a_3)+...(a_{n+1}+{a_n})$ Then $$2S_n=1+(-1+1)+(1-1)+(-1+1)+...0$$ ...
0
votes
0answers
29 views

Convergence of a Sum

I need to show that the $\lim_{n\rightarrow \infty } \sum_{i=1}^{n} (\frac{p_{i}(1 - p_{i})}{n^{2}} )$ converges to $0$. Where the $p_{i}$'s are just constants. In particular, they are probabilities ...
1
vote
2answers
59 views

Summation up to $n$ terms : $\sum r\cdot (r+1)^2$

Summation up to $n$ terms : $$\sum_{r=1}^{n} r\cdot (r+1)^2$$ My attempt : $$\sum_{r=1}^{n} r\cdot (r+1)^2=\sum_{r=1}^{n} r^3+2\sum_{r=1}^{n} r^2+\sum_{r=1}^{n} r$$ $$\sum_{r=1}^{n} r\cdot ...
1
vote
3answers
67 views

Is there a difference between $S_n$ and $V_n$

$$S_n=10+10+10+...$$ $$V_n=22-12+42-32+62-52+...$$ Do they represent the same series ?
0
votes
1answer
19 views

Order of a finite sum

How to prove that the order of $$ \sum\limits_{k=0}^{n} k^{-d/2 + 1} e^{-\frac{n}{k}} $$ is $O(n^\frac{4-d}{2})$? I would like to bound the sum by integral, but the function is not monotonic.