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

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1answer
147 views

problem with recurrence relation for series solution for ODE

I have $$y''-xy'-y=0$$ and I'm trying to find the series solution around the ordinary point $x_0=1$. My last post I muscled through to the solution when the ordinary point was $x_0=0$, but this is ...
2
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3answers
117 views

Prove: The infinite series $x_n$ converges if and only if the infinite series $2^nx_{2^n}$ converges.

I am in desperate need for hints to get me in the right direction for this proof. Let $(x_n)_{n\in \mathbb {N}}$ be a monotone decreasing null sequence. Prove that: $$\sum_{n=1}^\infty x_n ...
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1answer
52 views

Convergence functions

Let X be a nonempty set. I define a convergence function on X to be a partial function from the set of all sequences in X, to X, that satisfies the five additional conditions: Every constant ...
3
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3answers
148 views

Question on Rudin sequences?

In baby Rudin, Rudin shows that $$\lim_{n \to \infty}\sqrt[n]{p} = 1.$$ In the proof of limit he tries to prove that the limit is $1$. So he takes $x_n = \sqrt[n]{p} - 1$. I have never noticed this ...
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2answers
80 views

Proving $\lim_{n \to\infty} \frac{1}{n^p}=0$ for $p > 0$?

I'm trying to prove 3.20a) from baby Rudin. We are dealing with sequences of real numbers. Theorem. $$\lim_{n \to {\infty}} \frac{1}{n^p} = 0; \hspace{30 pt}\mbox {$p > 0$}$$ Proof. ...
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3answers
71 views

Convergence using Root Test

Problem: test if the series converges$$\sum_{n=1}^ \infty \frac {(-2)^{n+1}} {n^{n+1}} $$ My approach: I see it is equal to $$\sum_{n=1}^ \infty \frac {(-2)^n} {n^n} \cdot \frac {-2} n$$, and ...
0
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2answers
301 views

Interval of convergence for $\sum_{n=1}^{\infty}9(-1)^nnx^n$

I need to find the interval and radius of convergence and I'm really confused with what I'm supposed to be doing. Here is the problem: $\sum_{n=1}^{\infty}9(-1)^nnx^n$ I then used the ratio test to ...
1
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1answer
50 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 ...
6
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2answers
186 views

$\mathop {\lim }\limits_{n \to \infty } {1 \over {\sqrt n }} \left({1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} +\cdots+{1 \over {\sqrt n }}\right)$ [duplicate]

$$\mathop {\lim }\limits_{n \to \infty } {1 \over {\sqrt n }} \left({1 \over {\sqrt 1 }} + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }}+\cdots+{1 \over {\sqrt n }}\right)$$ ( Without use of integrals ...
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1answer
62 views

A result about quadratic numbers

I'm stuck in the middle of an exercise about quadratic numbers. Let me quickly sum it up. Let $d$ be a positive integer that is not the square of any integer. Let $n \in \mathbb N^*.$ Prove the ...
4
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1answer
3k views

Series solution to $y''-xy'-y=0$

So I'm learning to solve ODE's with series on my own using Boyce and DiPrima and exercise #3 is irking me...just looking for power series solutions around the ordinary point... $$y''-xy'-y=0$$ So I ...
1
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1answer
430 views

Weird sequence of numbers

Supply the missing number in the following sequence: 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, ___, 100, 121, 10.000. I've spent like 1,5 hours on this weird ...
6
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2answers
178 views

Borel sum of $ 1!+2!+3!+… $

I know that the Borel sum of $ \sum_{n=0}^{\infty}(-1)^{n}n! $ is $ \int_{0}^{\infty} dx \frac{e^{-x}}{1+x} $ but what happens with the sum $ \sum_{n=0}^{\infty}n! $ the Borel sum should be $ ...
2
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2answers
266 views

Limit of a fraction of double factorials

How can we show that $\begin{align*} \lim_{n\rightarrow\infty} U_n = 0 \end{align*}$ where $\begin{align*} U_n = \frac{(n-1)!!}{n!!}=\frac{n-1}{n}\frac{n-3}{n-2}\frac{n-5}{n-4}\cdots \end{align*}$ ...
1
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1answer
67 views

$1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+\dots-{1\over 2012}+{1\over 2013}$

The sum $1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+\dots-{1\over 2012}+{1\over 2013}$ is equal, a) ${1\over 1006}+{1\over 1007}+{1\over 1008}+\dots+{1\over 2013}$ b) ${1\over ...
0
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0answers
37 views

Recurrence in two variables

Anyone know how to solve the following recurrence relation in two variables: $$ f(x,y) = b f(x-1,y) + c f(y,x-1), \qquad \begin{cases}f(x,0) = b^{(x-1)} \\ f(0,y) = 0 \end{cases} $$ (Note: repost of ...
3
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1answer
146 views

Sum of reciprocal of zeroes of zeta function

The sum of reciprocal of zeroes of riemann zeta function converges conditionally that if they are paired as $\rho $ and $1-\rho$ My question is if the sum still converges if they are paired as $\rho$ ...
2
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5answers
170 views

How to find the limit of $\frac{\ln(n+1)}{\sqrt{n}}$ as $n\to\infty$?

I'm working on finding whether sequences converge or diverge. If it converges, I need to find where it converges to. From my understanding, to find whether a sequence converges, I simply have to find ...
1
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1answer
146 views

Probability of index at which sequence stops decreasing

Let $X_1,X_2, \dots $ be a sequence of independent and identically distributed continuous random variables. Let $N \ge 2$ be such that $X_1 \ge X_2 \ge X_{N-1} < X_N$. That is, $N$ is the ...
3
votes
1answer
93 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
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3answers
70 views

The definition of a sequence converging.

The sequence $\{a_n\}$ converges to t he number L if for every positive number $\epsilon$ there corresponds an integer $N$ such that for all $n$, $$n > N \implies |a_n - L| < \epsilon$$ If no ...
2
votes
2answers
126 views

What, if anything, does it mean to be neither finite nor infinite for real numbers?

I have this book which is said to be notoriously bad by my professor and the graduate T.A's and in a section titled "Series with Nonnegative Terms" the following statement appears: Every series ...
1
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1answer
50 views

What is $x$ if $\lim_{n\to\infty}{a_n} = 2$ while $a_1=x$ and $a_{n+1}=x^{a_n}$

I was told to find $x$ if I know ($a_n$)'s limit while $a_1=x$ and $a_{n+1}=x^a_n$, specifically for $\lim_{n\to\infty}{a_n}=2$ and then for different limits. It is pretty clear that $x=\sqrt{2}$, but ...
0
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2answers
91 views

Question about arithmetic–geometric mean

We have two sequences: $$a_{n+1}=\sqrt{a_nb_n}$$ $$b_{n+1}=\frac{a_n+b_n}{2}$$ I need to prove that those are making Cantor's Lemma.(At the end I shold get that: $\lim_{n\to \infty}a_n=\lim_{n\to ...
1
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3answers
139 views

Does $\sum_{n = 1}^\infty\frac{\ln^2(n)}{\sqrt{n}(8n + 9\sqrt{n})}$ Converge?

I've been working with the following series: $$\sum_{n=1}^\infty\frac{\ln^2(n)}{\sqrt{n}(8n+9\sqrt{n})}$$ I know that I must use the comparison test for convergence, but I'm unsure what to compare ...
3
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2answers
101 views

Limit $\mathop {\lim }\limits_{n \to \infty } n({1 \over {{{(n + 1)}^2}}} + {1 \over {{{(n + 2)}^2}}} + \cdots{1 \over {{{(2n)}^2}}})$

Without using integrals, how to find this limit: $$\mathop {\lim }\limits_{n \to \infty } {a_n} = n\cdot\left({1 \over {{{(n + 1)}^2}}} + {1 \over {{{(n + 2)}^2}}} + \cdots{1 \over ...
0
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1answer
83 views

Prove the convergence of sequences $x_0> 1, x_{n +1} = \log_2 (1 + x_n), n\geq 0$ and

To show $x_n, y_n$ defined by $$x_0> 1, x_{n +1} = \log_2 (1 + x_n), n\geq 0$$ and $$y_n =\frac{(1+x_1)(1+x_2)...(1+x_n)}{2^n}, n\geq1$$ are converging. For $x_n$, using inequality $2^x>1+x, ...
1
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1answer
337 views

Prove that the sequence of L-Lipschitz functions converge

$f_n(x): [a,b] \to \mathbb R$ are a sequence of functions that all are $L$-Lipschitz: means - $|f_n(x)-f_n(y)| \le L|x-y|$ , ($L$ is for all the functions) and assume $f_n \to f$ in a pointwise ...
0
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1answer
62 views

'ϵ-δ' proof for the following sequence

I need help writing a formal 'ϵ-δ' proof for the following sequence: $$ \lim_{n\to \infty}(n+2)^2 \sin(1/n)=\infty $$ Thanks in advance.
1
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2answers
78 views

Prove that the series converges to the integral

Prove: $\int _0^{1}x^{-x}dx$ = $\sum_{n=1}^\infty\frac{1}{n^n} $ I thought of using: $x^{-x}$ = $e^{-x lnx}$ and then using : $e^{-xlnx}$ = $\sum_{n=1}^\infty\frac{(-xlnx)^n}{n!} $ but I'm stuck from ...
1
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1answer
176 views

Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $a_n+b_n$? [duplicate]

Suppose $a_n$ converges and $b_n$ divergent. What can you tell about $(a_n+b_n)$? I know that if $(x_n)$ and $(y_n)$ converge, then both $(x_n+y_n)$ and $(x_n−y_n)$ converge also. But can't ...
1
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1answer
103 views

Got stuck with the leibniz criterion

$\sum_{n=1}^{\infty} \Big(\frac{1}{n^2}+ \frac{(-1)^n}{n} \Big)$ Does the progression converge, absolutely converge or diverge. I tried it with the Leibniz-criterion, but I dont know how to proof ...
3
votes
1answer
174 views

Sum of Infinite Series $1 + 1/2 + 1/4 + 1/16 + \cdots$

Everyone knows about the classic $$ \sum_{i=1}^{\infty} \dfrac{1}{2^i} = 1 $$ However, is there any way to find $$ \sum_{i=0}^{\infty} \dfrac{1}{2^{2^i}} = \dfrac12 + \dfrac14 + \dfrac{1}{16} + ...
2
votes
1answer
66 views

Limit of a sum of roots proof

Given the sequence: $$a_n=\alpha\sqrt{n+a}+\beta\sqrt{n+b}\ with\ \ \alpha,\beta,a,b\in\mathbb{R}\ and\ \alpha,\beta\neq0$$ Prove that $$\lim_{ n\to \infty} a_n = 0\ iff\ \alpha=-\beta$$ I start ...
2
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2answers
60 views

Dual of a sequence

Let $S$ be the set of all sequences $(a_1,a_2,\ldots)$ of non-negative integers such that (i) $a_1 \ge a_2 \ge \ldots;$ and (ii) there exists a positive integer $N$ such that $a_n=0$ for all $n \ge ...
2
votes
1answer
94 views

sum of an alternating series

How to evaluate the series below ? $$ \sum_{n=0}^{\infty}\left(-1\right)^{n}\,{2n+1 \over \left(2n+1\right)^{2} + x^{2}} $$ Can we reexpress it in term of an elementary function ?.$\,$ By the way, ...
2
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2answers
75 views

The series of reciprocals of the integers that do not contain 9 in their decimal representation

Does the following series converge or diverge? $\sum_{n=1}^{\infty} a_n$ where $a_n = \frac 1 b_n$, and $(b_n)_n$ is the subsequence of $(n)_n$ whose terms do not have a $9$ in their decimal ...
5
votes
4answers
99 views

Convergence of $\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$

I have to show that the following series convergences: $$\sum_{n=0}^{\infty}(-1)^n \frac{2+(-1)^n}{n+1}$$ I have tried the following: The alternating series test cannot be applied, since ...
1
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1answer
43 views

Formula for the terms of the sequence defined by $a_0 = 1$, $a_1 = -2$ and $a_{n}=-4 a_{n-1}-4 a_{n-2}$

Let $a_{n}$ be the sequence recursively defined by $a_{0} = 1$, $a_{1} = -2$, and for $n\geq 2$, $a_{n}=-4 a_{n-1}-4 a_{n-2}$. Use strong induction to show that $a_{n}$ = $(-2)^n$ for all n. ...
1
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1answer
56 views

Average of sequence random variables

Let $X_1, X_2, X_3, \dots$ be a sequence of random variables that converges almost surely $$(X_n) \rightarrow X$$ to a number $X \in \mathbb R$ (or more precisely the delta dirac distribution ...
1
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2answers
275 views

How to prove, by induction, that an infinitely nested radical is increasing

How do I prove using induction that an infinitely nested radical, like sqrt(1+sqrt(1+sqrt(1+... is increasing. I have seen there are many examples on here like this but haven't seen one that proves ...
4
votes
2answers
145 views

Find $L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$

Find the limit following: $$L=\lim \limits_{n\to \infty}\sqrt[n]{x^n+y^n+z^n}$$ With $x,\: y\: z\in R$ P.S I think this limit result is $L=max\left\{x,\: y\: z \right\}$. But i'm not find it, so ...
1
vote
1answer
340 views

Function iteration and intervals of attraction for fixed points

I am currently studying iteration sequences and I am a bit hung up on one specific bit which involves determining intervals of attraction of fixed points. I've been given a graphical method to ...
2
votes
2answers
71 views

Convergence: $\sum\frac1{n\ln(n^3)}$

How do you test the convergence of $$\sum_{n=2}^\infty\frac1{n\ln(n^3)}$$ I tried using limit comparison test, but had no conclusion.
1
vote
1answer
79 views

Sequence with a contraction mapping of the sum

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - ...
1
vote
2answers
60 views

Recursive sequence of functions

Recursive sequence of functions: $f_{n+1}= \sqrt{x+f_n}$, $f_1(x)= \sqrt{x}$. this sequence is monotonic, but what is bounding it? thanks
29
votes
1answer
669 views

Find the limit $L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$

Find the limit following: $$L=\lim_{n\to \infty} \sqrt{\frac{1}{2}+\sqrt[3]{\frac{1}{3}+\cdots+\sqrt[n]{\frac{1}{n}}}}$$ P.S I tried find this limit, but it's made me confused.
1
vote
1answer
134 views

Minimum of a sum

I have the function $$f(x)= \sum_{i=1}^n (x-a_i)^2 \ x\in R$$ I am asked to find the minimum of it. I am lost so any help would be nice. Thanks in advance!
2
votes
1answer
103 views

Let $\{a_n\}$ be a positive monotonic decreasing sequence of real numbers. Show $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$

Let $\{a_n\}_{n=1}^{\infty}$ be a monotonic decreasing sequence of positive real numbers. Show that $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$ Suppose we take the sum of the ...
4
votes
4answers
126 views

Evaluate $\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$

Question is to Evaluate : $$\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$$ what all i could do is : ...