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

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2
votes
1answer
114 views

Uniform convergence of $ x\arctan(nx)$

Let $f_n(x)=x\arctan(nx)$ for $n\geq1$. Show that $(f_n(x))$ converges uniformly to a function $f$. In the previous parts I have proved that $f_n(x)$ continuously differentiable and it converges ...
1
vote
0answers
77 views

What are the next few “tetranacci-like” pseudoprimes?

Starting with $n=0$: $k=2$ Given the roots $x_i$ of $x^2-x-1=0$. Then, we have the Lucas numbers, $$A_n = x_1^n+x_2^n = 2, 1, 3, 4, 7, 11, 18,\dots$$ The $n$ that divides $A_n-1$ are all the ...
2
votes
1answer
59 views

For which $z \in \Bbb{C}$ does this series converge?

How can I determine for which $z\in \Bbb{C}$ the following series converges? $$\sum_{n=0}^{\infty} \frac{z^n}{1+z^{2n}}$$ I've tried the root and ratio test with no success.
0
votes
1answer
43 views

How to find this sum: $\sum_{n=1}^{\infty}\frac{H^{(r)}(n)}{2^n n}$

I am learning new sums and I would appreciate your hints about how to approach the following $$S(r)=\sum_{n=1}^{\infty}\frac{H^{(r)}(n)}{2^n n}$$ where $$H^{(r)}(n)=\sum_{j=1}^{n}\frac{1}{j^r}$$ is ...
2
votes
1answer
291 views

Verification of Solution for Walter Rudin Principles of Mathematical Analysis Exercise 20, Chapter 3

I have written an answer for the problem 20, chapter 3 of Walter Rudin's Principle of Mathematical Analysis. I think the proof is correct, but since I am new with this kind of proofs, I am skeptical ...
1
vote
0answers
28 views

how to integrate this equation with two sums inside?

i'm reading a book, and i have trouble with this problem, i don't how to integrate this equation and where to begin from. Although they already give the answer but I don't understand how to get it. I ...
1
vote
1answer
35 views

prove or disprove this theorem about subsequences

I need to prove or disprove that for a certain sequence $a_n$, if the subsequence of the even indices and the subsequence of the odd indices are Cauchy sequences then $a_n$ converges.
3
votes
2answers
72 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 $\sum_{n=1}^{\...
0
votes
0answers
38 views

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

Prove the limit $\lim\limits_{n \to \infty}a_n$ exists and evaluate it for a recursive sequence $a_{n+2}=f_{(n\pmod 3)}(a_{n+1},a_n)$ where: $f_{0}(x,y)={1 \over 2}(x+y)$ $f_{1}(x,y)=\sqrt{xy}$ $f_{...
2
votes
4answers
101 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 \sqrt[p]{a_1.....
1
vote
2answers
51 views

How to prove this statement using Cauchy's 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.It is also known that ...
3
votes
4answers
449 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 it....
1
vote
2answers
1k 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
votes
3answers
88 views

Sum of series $\sin \theta+\sin 2 \theta+\sin 3\theta+\dots$ [duplicate]

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
22 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 - $a_{n+2}=\sqrt{a_{n+1}...
2
votes
3answers
76 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
55 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
124 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
128 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
117 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
102 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. \...
1
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0answers
38 views

A special limit-sum interversion

Let $l\in\mathbb{R}$ and $(u_k)_{k\in\mathbb{N}}$ a sequence converging to $l$. Let $a_{n,k\in\mathbb{N}}$ be such that $\displaystyle\forall n\in\mathbb{N},\sum_{k=1}^na_{n,k}=1$. $\forall k\in\...
5
votes
1answer
69 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
42 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
33 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?
3
votes
3answers
263 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 ...
3
votes
1answer
113 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 ...
2
votes
1answer
45 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 $||v||_\infty<\infty$. \begin{...
2
votes
2answers
192 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 \lim_{m\to\infty}f(n,m)...
0
votes
3answers
90 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
2answers
31 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
72 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
29 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
vote
5answers
181 views

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

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 ...
1
vote
1answer
29 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 $\left\{\frac{2n-1}{n}\right\}_{...
-1
votes
2answers
61 views

Find x in the following equation: [closed]

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.
2
votes
5answers
74 views

Prove that $a_{n+1}=a_{n} + 1/(n+1)^2$ is bounded from above

Prove that $a_{n+1}=a_{n} + 1/(n+1)^2$ is bounded from above What I have tried is $$a_n=1+1/4+\dots+1/n^2\leq 1+1+\dots +1=n$$ So I conclude that $a_n$ is bounded above by $n$. Does this ...
0
votes
2answers
44 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
430 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
109 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 $n ...
4
votes
2answers
938 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
37 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 $f$, i.e....
1
vote
1answer
63 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
62 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 ...
1
vote
0answers
26 views

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

Say we 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 $\sum_{n=0}^{\infty} (2n+1)t^n$. You can represent this as a ...
-2
votes
1answer
91 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
44 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
59 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
35 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 ...