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

learn more… | top users | synonyms (5)

1
vote
2answers
30 views

Cluster points and the sequence 1,1,2,1,2,3,1,2,3,4,1,…

I am working on a problem in analysis. We are given a sequence $x_n$ of real numbers. Then a definition: A point $c \in \mathbb{R}\cup{\{\infty, -\infty}\}$ is a cluster point of $x_n$ if there is a ...
0
votes
1answer
12 views

Geometric progression with reverse order

I have the following problem: Find three positive numbers which have the sum of $70$ and create a Geometric progression ($q>0$, increasing). Their inverse sum equals to $4/70$. Thank you!
1
vote
1answer
40 views

Proving non-repitition of a sequence

I have heard that the sequence $$x_{n+1}=rx_n(1-x_n)$$ for $r$ between $3$ and $4$ does not recur i.e. there is no $a>0$ such that $x_{n+a} = x_n$ and $x_0$ is any number between 0 and 1 exclusive. ...
4
votes
2answers
39 views

Sequence bounded away from $0$ and $2$

Suppose I have a sequence of real numbers $\{a_n\}_n$ and I'm told that $\{a_n\}_n$ is bounded away from $0$ and $2$. (1) What does it mean exactly? My thinking is that it means $a_n\neq 0$ and $a_n ...
3
votes
2answers
169 views

Applications of Dominated/Monotone convergence theorem

Consider a measure $\mu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ where $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$-algebra on $\mathbb{R}$. Consider the function $f: [0,\infty)\rightarrow ...
1
vote
0answers
22 views

Denominators of harmonic numbers: asymptotic behaviour.

About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts: It is unbounded, since $p\mid d_p$ for any prime $p$. It contains only one $1$. What more is known? Specially, ...
0
votes
1answer
34 views

Given series diverges or converges?

Given: $$\sum_{i=1}^\infty \frac2{7*i + 21} $$ The limit of the $nth$ term is 0, it means we aren't sure if it diverges. On wolfram it says it diverges by comparison test, but how?
0
votes
1answer
50 views

Maximum value of $\lambda$

It is given that a,b,c are be of same sign and a,b,c are in Harmonic progression i.e. $\frac{2}{b}=\frac{1}{a}+\frac{1}{c}$ and also $\frac{a+b}{2a-b}+\frac{c+b}{2c-b}\geq \sqrt{\lambda \sqrt{\lambda ...
0
votes
3answers
21 views

Prove the existence of limit of certain sequences.

Problem: Let $0<a_1<b_1$ and $$a_{n+1}=\sqrt{a_n\cdot b_n},b_{n+1}=\frac{a_n+b_n}{2}.$$ Prove that $\{a_n\}$ and $\{b_n\}$ converge to some limit. Attempt: By induction and AM-GM, I can show ...
1
vote
1answer
31 views

Limit of Variant of Geometric Sequence with Common Ratio any Real Number Between 0 and 1

I was wondering if the result that the limit of a geometric sequence with common ratio $r\in(0,1)$ is 0 held if successive terms were multiplied by any real number between $0$ and $1$. In a recursion ...
3
votes
1answer
57 views

Prove if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, {$b_n$} is bounded & monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges.

Prove that if $\displaystyle \sum_{n=1}^ \infty a_n$ converges, and {$b_n$} is bounded and monotone, then $\displaystyle \sum_{n=1}^ \infty a_nb_n$ converges. No, $a_n, b_n$ are not necessarily ...
0
votes
1answer
20 views

Calculus Series and simplifying the expression

Considering the following series, Series How do I find a simplified expression for the ratio (an+1 / an)
3
votes
1answer
74 views

Is there a name for an infinite product series?

I already know about the Harmonic series: $$\sum_{n = 1}^{\infty} \frac 1n = 1 + \frac 12 + \frac 13 + \frac 14 + \frac 15 + \frac 16 + \cdot \cdot \cdot$$ But is there a name for this infinite ...
0
votes
1answer
235 views

Is there an integral or series for $\frac{\pi}{3}-1-\frac{1}{15\sqrt{2}}$?

The approximation $$\pi\approx\frac{22}{7}=3+\frac{1}{7}$$ suggests that the closest integer to $\frac{1}{\left(\pi-3\right)}$ is $7$. However, $$ \frac{1}{\left(\pi-3\right)^2}\approx49.879 $$ is ...
-2
votes
2answers
67 views

Cant find the pattern [closed]

These are two sequences: I tried looking for a ratio or difference but it wasnt working $$ 12,20,32,\underline{\quad},22,\underline{\quad},50 $$ $$ 6,10,16,8,\underline{\quad},22\underline{\quad} $$ ...
0
votes
0answers
35 views

Show that $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ and $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$

Could you help me to show that (1) $\lim_{n\rightarrow \infty} (1-\omega(\frac{1}{n}))^n=0$ (2) $\lim_{n\rightarrow \infty} (1-o(\frac{1}{n}))^n=1$ where $o(\cdot)$ is little $o$ notation described ...
0
votes
1answer
24 views

Limit of a function defined as the sum of a series

Given a decreasing sequence $a_n$ of positive real numbers, for $x>0$ define $$ f(x)=\sum_n \min\left(\frac{x}{a_n}, \frac{a_n}{x}\right). $$ Can $a_n$ be chosen so that $f(x)\to 0$ as $x\to 0$?
1
vote
4answers
76 views

can't determine the convergence/divergence here

Let $$t_{n}=\frac{1}{n}\left(1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}\right),\ n=1,2,\dots$$ then I want to know if $\sum_{n=1}^{\infty}t_{n}$ converges/diverges and the sequence$\{t_{n}\}$ ...
-2
votes
0answers
42 views

What is the sum of the series: $1+2(1/3)+3(1/3)^2+4(1/3)^3+\cdots$ [duplicate]

Please help me find the sum of this series. I tried separating the geometric series pattern from it but the remaining part is just the same.
0
votes
1answer
29 views

Convergence of the sequence $z_1=\frac{3}{2}$ with $z_{n+1}=\sqrt{3z_n-2}$

Prove the convergence of the sequence $(z_n)$ such that : $$ z_1=\frac{3}{2}$$ $$z_{n}=\sqrt{3z_{n-1}-2}$$ for every $ n \geq 2$. Calculate also the limit. I have applied induction: $$\frac{3}{2} ...
0
votes
2answers
26 views

Question concerning limit superior and inferior.

Let ${a_n}$ and ${b_n}$ be two real sequences such that $a_n\leq b_n$ for all $n$. Is it true that $\lim \sup a_n \leq \lim\inf b_n $? Outline the proof if so.
-1
votes
0answers
60 views

Proof of 1^2 + 2^2 + 3^2 + 4+2 +… infinity = 0 [closed]

Leonhard Euler proved that the sum of the series $$ 1^n + 2^n + 3^n + 4^n + ... ∞ = 0 $$ where n is an even number. I know that this is a diverging series, but the infinity part of it somehow makes ...
17
votes
1answer
1k views

Sum equals integral

It is quite a well known fact that: $$\int_0^{+\infty} \frac{\sin x}{x} \, dx = \frac{\pi}{2}$$ also the value of related series is very similiar: $$\sum_{n = 1}^{+\infty} \frac{\sin n}{n} = \frac{\pi ...
1
vote
2answers
28 views

find the limits of…

$$A_n=\frac{cos(n)}{n}\to 0$$ I've been asked to find the limits of this sequence however using the sandwich theorem I have just found that the limit is $0$ is this the only limit if not what else is ...
2
votes
2answers
49 views

Solving the recurrence $A_n = \sum_{k=1}^{n} 2^{k+1} A_{n-k}$

Let me ask a very simple question: Let $(A_n)$ be a sequence of integers defined by $A_0 = 1$ and $$\forall n \geq 1 : A_n = \sum_{k=1}^{n} 2^{k+1} \cdot A_{n-k}.$$ There is an explicit formula for ...
3
votes
2answers
375 views

Question on Riemann's rearrangement theorem

Let there be given some conditionally convergent infinite series $S$. Then let $R$ be some real number, and $Q_k$ a rearrangement of $S$ such that the sum is equal to $R$. Is $Q_k$ unique? In other ...
0
votes
0answers
10 views

Smoothing a function by subtracting terms in its Taylor series?

I am looking at some code for a Greens function that mentions the following % The GF is the smoothed by subtraction of first two odd Taylor series terms. So how does subtracting terms from a ...
0
votes
2answers
31 views

What do we mean by convergence of a series?

While learning calculus I stumbled upon this concept of convergence. Is this some general concept or just related to sequence and series. What is its importance?
0
votes
2answers
30 views

How do I evaluate this sum :$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$?

I'm interesting to know how do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{{(-1)}^{n²}}{{(i\pi)}^{n}}$$, I have tried to evaluate it using two partial sum for odd integer $n$ and even integer $n$ ...
2
votes
2answers
69 views

Compute $(\sin4^\circ)^2 +(\sin8^\circ)^2+(\sin12^\circ)^2+\cdots+(\sin176^\circ)^2$

Angle of sine is in degrees, can anyone show me an easy soln to this? This was question was given to us for 1minute without calcu. I know that $\sin4^\circ=\sin176^\circ$, ...
0
votes
4answers
94 views

In need of assistance with evaluation of a tricky limit [closed]

I am trying to evaluate a limit, but without much luck. I keep getting infinity as the answer, but both Wolfram and my textbook state otherwise. The limit problem goes as follows: $\lim_{n \to ...
0
votes
1answer
26 views

For a sequence, why must $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {||x_n||} = 0$, or there exists a convergent subsequence with a nonzero limit?

Suppose I've got a sequence of vectors $\{x_n\}_{n∈N}$ in $\mathbb{R}^k$. Why is it that exactly one of the following three facts must hold: $\lim _{n→∞} {||x_n||} = ∞$, $\lim _{n→∞} {x_n} = 0$, or ...
1
vote
3answers
54 views

How does the minus come in Geometric Series

I'm looking into the geometric series and can't understand how the 1 - .01 comes in below: ...
2
votes
0answers
32 views

For x between 0 and 1, what does the following series converge to?

Consider the following infinite series: $\sum_{i = 1}^{\infty} 2^i x^{(2^i)} = 2x^2 + 4x^4 + 8x^8 + 16x^{16} + ...$ I know that it converges for values of x between 0 and 1- can anyone help find a ...
3
votes
0answers
59 views

Prove $ \limsup a_n$ is a real number

($a_n$) is a real sequence bounded only from above. Let $S :=$ {$t \in \Bbb R:$ $t$ is the limit of a convergent subsequence of ($a_n$) }. Suppose the supremum of S is a real number. Prove that $ ...
1
vote
0answers
14 views

Ratios between index and sequence element

Let $a_1\le a_2\leq\ldots$ be an infinite sequence of positive integers. A positive integer $n$ is called good if $i=na_i$ for some index $i$. For which $(m,n)$ is it true that if $m$ is good, then ...
2
votes
1answer
55 views

Show $A=\limsup_\limits{n\to\infty}a_n$.

Let $\{a_n\}$ be a sequence of real numbers bounded from above, $A\in \Bbb R$. Given any $\epsilon>0$, a)$\exists n_0 \in \Bbb N$ such that $ a_n<A+\epsilon$ for all $n\ge n_0$. b)$\exists ...
11
votes
2answers
253 views

Is there an algorithm to determine whether a series converges?

I thought of this question in connection with Calculus II (a course in the US which includes, among other things, techniques of integration and convergence tests for series, both of which are taught ...
2
votes
1answer
45 views

Summation of Hardy's series

I recently found this series from an Hardy work: $$\sum_{n=0}^{+\infty}(-1)^nx^{2^n}=x-x^2+x^4-x^8+\dots$$ For what values of $x$ does it converge ? Can we use some summation technique to sum it ...
1
vote
0answers
9 views

Monotonically decreasing sequence/series proof [duplicate]

I have a proof that I'm working on and it goes like this: Assume $a_k > 0$ and $a_k$ is monotonically decreasing. Show that: $$\sum_{k=1}^\infty a_k < \infty \iff \sum_{k=0}^\infty b_k < ...
4
votes
4answers
106 views

Prove: $\operatorname{E}[X^2]<\infty\Longrightarrow \operatorname{E}[X]$ exists

I don't know how to prove this. Lets assume that $X$ is a discrete random variable. I've just come this far: If we do a direct proof of the implication, then we start with the assumption: $$ ...
-1
votes
1answer
36 views

Regarding infinite geometric series, how to derive the value for r.

This question will only require a simple explanation, however thus far I have not been able find this from my notes or the web. The sum of an infinite series is: $$a/(n)r$$ if the series is ...
0
votes
2answers
38 views

Meaning of $\lim_{n \rightarrow \infty } \sup$ $ a_n$

Consider a real sequence $\{ a_n \}_{n \in \mathbf{N}}$. Now if I write $A = \lim_{n \rightarrow \infty }$ $ \sup$ $ a_n$ , what do I actually mean by this? I mean if $a_1 , a_2, . . . $ is my ...
0
votes
1answer
49 views

Accelerated Order of Convergence

Let $m > 0$ and $ a:[0,1] \rightarrow \mathbb R$ be a function with $a(\epsilon) \rightarrow 0$. Then $ \epsilon^m a(\epsilon) \rightarrow 0 $ for $ \epsilon \rightarrow 0$. But is $ \epsilon^m ...
0
votes
1answer
43 views

Prove a square free word

A square free word is a word that does not contain any subword twice in a row. An infinite word is defined: $$ w_i=\{ \textrm{the maximal natural j that } 2^j \textrm{ devides } i\} $$ the first ...
3
votes
2answers
53 views

If $\lim_{n \rightarrow \infty} a_n=L$ then $\lim_{n \rightarrow \infty} f(a_n)=f(L)$?

If we have for example $a_n=1+\sqrt{a_{n-1}}$ and $\lim_{n \rightarrow \infty} a_n=L$ then can I say that $ L=1+\sqrt{L}$? If it's so, what's the proof?
2
votes
5answers
79 views

Limit of the sequence defined by a recurrence

Given a recurrence formula for an arithmetic sequence, $$U_{n} = \frac{1}{2+U_{n-1}}$$ Show that$$\frac{1}{2+\frac{1}{2+\frac{1}{2+\frac{1}{2+ ...}}}} = (SomeGivenValue)$$ How can we solve questions ...
1
vote
3answers
36 views

Show that the sequence $\left(\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(0,1)$ decreases monotonically and converges to $0$

I have to show that sequences $\left(\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(0,1)$ and $\left(-\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(-1,0)$ decrease monotonically and converge to $0$. I ...
3
votes
3answers
120 views

Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$

I want to show that for all $a \in \mathbb{R }$ $$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$ So far I've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i ...
32
votes
25answers
27k views

Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$. How? What's the proof? Or maybe it is self apparent just looking at the above? PS: This problem is know as "The sum of the first $n$ positive ...