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

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2
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2answers
25 views

Interesting fact about convergent sequences?

Let $a_n > 0$ for $n=1,2,...,$ with $\sum_{n=1}^{\infty}a_n < \infty$. Prove that $b_n$ $(n=1,2,...)$ exist such that $b_n/a_n \rightarrow \infty$ as $n \rightarrow \infty$, but ...
0
votes
1answer
14 views

Evaluate $\lim_{n \to \infty}a_n$ for a recursive sequence $a_{n+2}=f_{n mod 3}(a_{n+1},a_n)$ where $f_{k}(x,y)$ is the mean of $x,y$.

Prove the limit $\lim_{n \to \infty}a_n$ exists and evaluate it for a recursive sequence $a_{n+2}=f_{n mod 3}(a_{n+1},a_n)$ where: $f_{0}(x,y)={1 \over 2}(x+y)$ $f_{1}x,y)=\sqrt{xy}$ $f_{2}(x,y)={2 ...
1
vote
3answers
40 views

The convergence of the arithmetic mean into the geometric mean

Given $p$ positives values $a_1...a_p$, define the sequence $x_n$ such that: $$x_n = \frac{\sqrt[n]{a_1}+...+\sqrt[n]{a_p}}{p}$$ And define $S_n = (x_n)^n$ Prove that $S_n \rightarrow ...
1
vote
1answer
34 views

How to prove this statement using Cauchys statement?

if $b_n>0$ is a sequence which implies that for every $\epsilon>0$ exists a certain $N$ so that for $m>n>N$ the expression $\sum_{k=n}^m b_k$< $\epsilon$ is true.How to prove the last ...
3
votes
4answers
94 views

Recursive sequence with square root

I came across this (cool) question this weekend Find the limit of the following sequence as $n$ approaches infinity. $x_1 = 1$ and $x_{n+1} = \sqrt{x_n^2+\frac{1}{2}^n}$ I had two questions about ...
0
votes
2answers
22 views

Prove that every convergent sequence has a monotone subsequence [duplicate]

So if a certain sequence $a_n$ is convergent then its bounded.So from Bolzano-Weierstrass $a_n$ has a convergent sub-sequence, but where do I continue from here?
-1
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3answers
52 views

Sum of series $\sin \theta+\sin 2 \theta+\sin 3\theta+\dots$

I need to prove sum of series: $$\sin \theta+\sin 2 \theta+\sin 3\theta+\dots=\sum_{n=1}^\infty \sin n\theta$$ by using in the first place the complex numbers.
0
votes
1answer
9 views

The limits of recursive sequences of different types of means - my solution + challenge

Consider the sequence $a_{n+2}=f(a_1,a_2)$ where $f(x,y)$ is the mean of $x, y$ (geometric/arithmetic/harmonic) and $a_1,a_2$ are positive real numbers. In detail: Geometric - ...
2
votes
3answers
47 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...
0
votes
1answer
30 views

$\lim_{n\to\infty}\sup x_n= \max(x,y)$ and $\lim_{n\to\infty}\inf x_n= \min(x,y)$

I am trying to prove this: Let $x_n$ a real sequence. Suppose that the subsequence $x_{2n}$ converges to $x$ and $x_{2n+1}$ converge to $y$. Show that: $\lim\sup_{n\to\infty} x_n= \max(x,y)$ and ...
1
vote
4answers
107 views

Series behavior using the Ratio Test

My professor gave us some food for thought today. A classmate asked a question and the professor didn't answer it but instead asked us to think about it. We must find a series $A_n$ with $A_n \geq 0$ ...
0
votes
1answer
19 views

Uniform convergence of $\{\tanh(nx)\}_{n=0}^{\infty}$

Quick question. How can I prove that the sequence of functions: \begin{equation} f_n(x)=\{\tanh(nx)\}_{n=0}^{\infty} \end{equation} converges uniformly to: \begin{equation} f(x)=\begin{cases} -1, ...
1
vote
2answers
36 views

Prove that the sum of the first $n$ even terms of the Fibonacci sequence is given by $(F_{3n} + F_{3n + 3} - 2) / 4$ not using induction?

each even term in the Fibonacci sequence has a position index which is a multiple of 3, therefore the even term is $F_{3n}$: $F_{3 (0)} = 0$ $F_{3 (1)} = 2$ $F_{3 (2)} = 8$ $F_{3 (3)} = 34$ $F_{3 ...
0
votes
0answers
15 views

Use a sequential argument to show that if $c \neq 0, f$ does not have a limit at $c$.

Here, $f$ is a real valued function given by $f= x, x\in \mathbb Q$ $=0, x\in \mathbb {R-Q} $ I'm trying to prove this by using the sequential divergence criterion. Let $c\neq 0 \in \mathbb Q. ...
5
votes
1answer
49 views

Compute $\lim\limits_{n\to \infty}\frac{\prod\limits_{k=1}^{n}a_k}{2^n}$

Compute $$\lim\limits_{n\to\infty}\frac{\prod\limits_{k=1}^{n}a_k}{2^n}$$ where $a_k=\sqrt{2+a_{k-1}}$ and $a_1=\sqrt{2}$. I proved $\lim\limits_{n\to\infty}a_n$ exists and found it (it's 2), but ...
0
votes
2answers
28 views

Direct Comparison Test

I have the equation $$\sum_{n=1}^\infty\frac{(-1)^nn\sqrt{n+1}}{\sqrt{n^3+2}} $$ After simplifying the numbers and using the direct comparison method. You end up with the following. So would you ...
0
votes
1answer
19 views

How to tell whether a complex sequence converges?

How do you tell whether a sequence with complex parts converges? For example, what would you do to prove whether the sequence $z_{n}=\frac{n}{(1+i)^{n}}$ converges?
2
votes
3answers
220 views

Prove if the following sequence is convergent

$$ f(n) =\frac{6n^3 + 2n+(−4)^n}{4^n-1} $$ The sequence is dominated by $4^n$, so we divide by the dominant term, and we get $(-1)^n$ which is not a null sequence. So it is not monotonic and not ...
1
vote
1answer
90 views

Compute $\frac{1}{e}\sum\limits_{n=0}^{\infty}\frac{n^{k}}{n!}$ for $k=0, 1, 2 … $

Using some matlab (I know it's cheating) I found that: $$k=0 => Result=1$$ $$k=1 => Result=1$$ $$k=2 => Result=2$$ $$k=3 => Result=5$$ $$k=4 => Result=15$$ $$k=5 => Result=52$$ $$k=6 ...
1
vote
1answer
28 views

The infinity norm of the sequence $v(n) = n \sin(n!)/(n^2+1)$

For a bounded sequence $v(n)$, $n\in\mathbb{Z}$ define $$||v||_\infty = \max_{n\in\mathbb{Z}} |v(n)|.$$ Let $$v(n) =\frac{n\sin{(n!)}}{n^2+1},$$ and find whether $||v||_\infty<\infty$. ...
2
votes
2answers
107 views

Changing limits in absolutely convergent series

Let $\sum_{n=0}^\infty f(n,m)$ be a real series. Suppose the series converges absolutely. Can we do the following? $$ \lim_{m\to\infty}\sum_{n=0}^\infty f(n,m)=\sum_{n=0}^\infty ...
0
votes
3answers
58 views

What does $\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} $ simplify to?

A solution to one of the exercises in my text states: $$\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} - \frac{1}{N} - \frac{1}{N+1}$$ I have no idea ...
0
votes
0answers
20 views

Show by comparison that the series $\sum_{n=2}^{\infty} \dfrac{1}{n^p (\ln(n))^q}$ is convergent if p>1 for all $q \in \mathbb{R}$ [duplicate]

So, I have to show (as the title says), that the series $\sum_{n=2}^{\infty} \dfrac{1}{n^p (\ln(n))^q}$ is convergent if p>1 for all $q \in \mathbb{R}$ by comparison. I've managed to show it for ...
0
votes
2answers
26 views

Series convergent mathematics question

I would be very grateful if someone could help me with this question on convergent series. I know how to answer the question but am stuck on the cancelling down of terms. I have attached a picture of ...
1
vote
1answer
54 views

Show that exist $i>0$ such that the Fibonacci number $F_{i}$ is divisible by 2015

This is a problem that has haunted me for more than a month. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind: Assume that the sequence ...
0
votes
3answers
19 views

Find a sequence (Sn) that for any a between [0,1] there is a subsequence of Sn that converges to a.

Find a sequence (Sn) that for any a between [0,1] there is a subsequence of Sn that converges to a. I've been stumped for days, my guess is that it is an addition of sequences each expressing its ...
-1
votes
5answers
91 views

Where is the mistake in proving 1+2+3+4+… = -1/12?

https://www.youtube.com/watch?v=w-I6XTVZXww#t=30 As I watched the video on YouTube of proving sum of $$1+2+3+4+\cdots= \frac{-1}{12}$$ Even we know that the series does not converge. First I still ...
0
votes
1answer
8 views

Prove that the sequence is Cauchy

This is problem 17, page 55, from section 1.2: Cauchy Sequences in the textbook Introduction to Analysis, Fifth Edition, by Edward D. Gaughan. Prove that the sequence ...
0
votes
2answers
47 views

Find x in the following equation: [on hold]

Find x in the following equation: $$\frac{x}{\sqrt{2}+1}+\frac{x}{\sqrt{3}+\sqrt{2}} +\frac{x}{\sqrt{4}+\sqrt{3}}+...+\frac{x}{\sqrt{2025}+\sqrt{2024}}=4004$$
0
votes
1answer
17 views

Given a summation figure out the alternating series

I figured out that the top is (2x-1) and that the difference between the denominator ends up being (2x-1), just not sure how to figure out what the series is.
0
votes
2answers
33 views

If $x_n$ $\rightarrow $ 1 Then show that sequence $\frac {4+ (x_n)^{2}}{x_n}$ approaches to limit 5

If $x_n$ $\rightarrow $ 1 Then show that sequence $\frac {4+ (x_n)^{2}}{x_n}$ approaches to limit 5 I have tried to find epsilon proof ,But i am not successful .Can anyone help me with this ...
0
votes
2answers
27 views

Prove that $(n+\frac {(-1)^n}{n}) $ is not Cauchy

Let $x_n=(n+\dfrac {(-1)^n}{n}) $ So, $|x_{2m}-x_{2n}|=|(2m+\dfrac {(-1)^{2m}}{2m})-(2n+\dfrac {(-1)^{2n}}{2n})|$ $=|2m-2n+\dfrac{1}{2m}-\dfrac{1}{2n}|$ $=|(m-n) (2-\dfrac{1}{2mn})|$ This is ...
1
vote
3answers
69 views

Show that the sequence defined recursively by $a_{n+1} = \sqrt{3a_n + 1}$ is increasing

Assume that the sequence ${a_n}$ is defined recursively by $a_{n+1} = \sqrt{3a_n + 1}$ for all $n \in \mathbb N$, with $a_1 = 1$. Use mathematical induction to prove that $a_n \leq a_{n+1}$ for all ...
4
votes
2answers
172 views

Find the sum of the following series 3 - 3/2 + 3/4 - 3/8 + 3/16 - 3/32 + …

The problem is an alternating series, that looks like this: I am given the series: The book mentions the Alternating Series Estimation Theory, however it seems like there is a definite answer by ...
1
vote
1answer
16 views

Question about the Range of a sequence

Consider a sequence $\{x_n\}$ in a metric space $(X,d_x)$. The sequence is a mapping from the natural numbers to $X$, $f:\mathbb{N} \to X$. We say that the range of a sequence is the range of $f$, ...
1
vote
1answer
46 views

telescoping series question?

Are all telescoping series absolutely convergent? if not, is there an example of a telescoping series that is not absolutely convergent?
0
votes
0answers
28 views

Proving continuity using Weierstrass M-test

I am asked to use the Weierstrass M-test to show that the following function is continuous on $A = \mathbb{R}\setminus \mathbb{Z}$ $$f(x) = \sum_{n=1}^\infty \frac{1}{x+n} + \frac{1}{x-n}$$ My ...
0
votes
0answers
5 views

Is the hilbert series of graded module interpreted as a rational function meaningful everywhere?

Say you have a graded module $M$ over a field $k$. In my mind I have $M=k[x,y,z]/(y^2-xz)$ graded by rank. The Hilbert series of this is $\sum_{n=0}^{\infty} (2n+1)x^n$. You can represent this as a ...
-2
votes
1answer
65 views

Proof that $\sum_{n=1}^\infty n $ is -1/12 [duplicate]

Why is the sum of all natural numbers $- \frac1{12}$? I need a proof my 14 year-old classmates could understand, with minimal effort on my part ;) I know I can prove it using zeta functions, etc. ...
0
votes
3answers
38 views

How do I calculate the limit of this integral from n to n+2?

I need to find the limit, as $n\to\infty$ of $\int_n^{n+2}e^{-x^3}dx$. I tried taking the integral using integration by parts but that doesn't work so now I'm stuck.
1
vote
1answer
53 views

Prove $\sum\limits_{1}^{\infty}(-1)^{n-1}\frac{2n+1}{n(n+1)}=1$

I found this in the beginning of a calculus book, so it should be solved with very basic techniques, but I really don't know how.
1
vote
2answers
26 views

How do I prove this with induction?

I am give $a_{n+1}=\sqrt{a_{n}+12}$ and $a_{n}∈[-12, 4]$. I need to prove $0≤a_{n}≤4$ for all $n≥2$. I have that $a_{2}∈[0,4]$ so it works for the first case and $a_{3}∈[\sqrt{12},4]$ so it holds for ...
8
votes
1answer
41 views

Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+\text{ such that } a^2-b=k^2 $?

This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: $$\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb ...
1
vote
5answers
34 views

Calculus 2 Series Convergence - Can I Use Comparison Test?

Can I use the comparison test for the following problem? $$\sum _{n=1}^{\infty }\:\frac{1}{n^2-6n+10}$$ The denominator has a negative coefficient so i'm not sure if its valid to compare it to a ...
1
vote
1answer
34 views

Prove that $f_n(x)=\frac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$

Title says it all; I have to prove that the function sequence $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$, with ...
1
vote
2answers
29 views

What are the rules of substituting in a power series?

I realize that to get the maclaurin of $e^{-2x^2}$ it is easiest to simply use the famous $e^x$ series and plug in $-2x^2$ for $x$ in all the terms. What are the limits of this? For example, could I ...
2
votes
1answer
40 views

Infinite sum of $\dfrac{1}{n^p \ln(n)^q}$

Let $p,q \in \mathbb{R}$. Show using the comparison test (or limit comparison test) that $$ \sum\limits_{n=2}^{\infty} \dfrac{1}{n^p \ln(n)^q} $$ converges for $p>1$ and any value of $q$ and ...
2
votes
1answer
17 views

Show that there exist $k$ and $r$ such that the given sum is divisible by $n$

Let $a_{1},\dots,a_{n}$ be integers. Show that there exist integers $k$ and $r$ such that the sum $$a_{k}+a_{k+1}+\dots+a_{k+r}$$ is divisible by $n$. I am unable to find the necessary way to solve ...
0
votes
1answer
27 views

Proof that limit of sequence is unique

I am learning real analysis on my own from this book http://books.google.co.in/books?id=TZ-NAgAAQBAJ&printsec=frontcover#v=onepage&q&f=false On page 33 , i do not get proof of that limit ...
3
votes
1answer
21 views

Inverse Rule for Formal Power Series

I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series: ...