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

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3
votes
3answers
1k views

Prove that the sequence $\cos(n\pi/3)$ does not converge

EDIT: Using a rigorous formal proof, I need to prove that this sequence does not diverge. I of course understand why it doesn't converge... $n=1$ to infinity of course. So, I have a bit of trouble ...
4
votes
1answer
66 views

Is it true for every sequence $a_n$ that if $\sum a_n$ is absolutely convergent, then $\sum (-1)^n a_n$ is convergent?

The problem is in the title. I must answer the question whether it's true for every sequence $\{a_n\}_{n\geq 1}$ that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent then $\sum_{n=1}^{\infty} ...
3
votes
2answers
290 views

Subsubsequence converges $\implies$ sequence converges

Prove that if $\left\{ x_n \right\}$ is an infinite sequence of real numbers, $x \in \mathbb{R}$, and every subsequence $\left\{ x_{n_k} \right\}$ has a subsequence $\left\{ x_{n_{k_j}} \right\}$ ...
0
votes
1answer
91 views

Prove the sequence $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge

How would I be able to prove that $(\cos(\frac{n\pi}{3}))_{n=1}^\infty$ does not converge? I know that for a sequence to converge to a limit, then for all $\varepsilon > 0, \exists N \in \mathrm N ...
4
votes
1answer
46 views

Prove $\{g(x_n)\}_{n=1}^\infty$ converges

Let $g : (a, b) → R$ be uniformly continuous on $(a, b)$. Let $\{x_n\}_{n=1}^\infty$ be a sequence in $(a, b)$ converging to $a$. Prove that $\{g(x_n)\}_{n=1}^\infty$ converges. The general idea here ...
4
votes
2answers
51 views

If $f_1(k)=\sum_{i=1}^k\frac{1}{i}$ and $f_n(k)=\sum_{i=1}^kf_{n-1}(i)$, then what is $f_n(n)$?

Let $$f_1(k)=\sum_{i=1}^k\frac{1}{i},$$ and define inductively $$f_n(k)=\sum_{i=1}^kf_{n-1}(i).$$ So, $$f_2(k)=\sum_{i_2=1}^k\sum_{i_1=1}^{i_2}\frac{1}{i_1},\quad ...
2
votes
1answer
73 views

Laurent series of $f(z)=\frac{1}{z(z-1)}$ given four different conditions

Expand $f(z)=\frac{1}{z(z-1)}$ in a Laurent series valid for the follwing annular domains. $a)0\lt \vert z \rvert \lt 1 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,b)1\le\lvert z ...
3
votes
2answers
48 views

Limit of a sequence containing root of n!-th degree - how to deal with that?

Here is a sequence the limit of which I'm trying to find as $n$ goes to infinity: $$a_n=\sqrt[n!]{\frac{1}{2^{n!}}-\frac{1}{3^{n!}}}$$ Here is what I've done: ...
2
votes
0answers
38 views

Function continuous at irrationals and discontinuous at rationals [duplicate]

Q: Given the function $f(x)=\sum_{n=1}^\infty f_n(x)$, where $f_n(x)= \left\{ \begin{array}{lr} 0; \;\;if \;x< r_n \\ \displaystyle \frac{1}{2^n}; x\geq r_n ...
2
votes
2answers
85 views

Find $\lim_{n \to \infty} \frac{(\sqrt[n]{(7^n+n)}-\frac{1}{7})^n}{7^n-n^7}$

Find $$\lim_{n \to \infty} \frac{(\sqrt[n]{(7^n+n)}-\frac{1}{7})^n}{7^n-n^7}$$ I know that it can be done with using the squeeze theorem but I cannot find a proper upper bound limit
0
votes
1answer
34 views

Sequence Recurence Relation problem help

$$a_1=1,\quad a_{n+1}=(a_n)^4.$$ Hi can someone help me with this sequence? thanks
0
votes
1answer
156 views

Convergence of $a_n = (1+i)^n+(1-i)^n$

I am asking a question related to Is there a formula for $(1+i)^n+(1-i)^n$? I am looking on the exact same term, just as a sequence, so i want to find out: Is $a_n = (1+i)^n+(1-i)^n$ convergent or ...
4
votes
7answers
672 views

Why does this sequence converge?

I have to deal with the following sequence : $\lim \limits_{x \to \infty}\sqrt{x+\sqrt{x}} - \sqrt{x}$ If I factorize it to $\sqrt{x}(\sqrt{\sqrt{x}+1}-1)$, I would say it diverges since both ...
1
vote
2answers
54 views

Why is $\sum \frac{\sqrt{n^3+n}-\sqrt{n^3}}{\sqrt{n(n+1)}-1}$ convergent?

We have the following series: $$\sum_{n=1}^{\infty} \frac{\sqrt{n^3+n}-\sqrt{n^3}}{\sqrt{n(n+1)}-1}$$ According to WolframAlpha it is convergent but I don't see why. Intuitively, the expression ...
3
votes
3answers
142 views

How to show that $\tan(n), n\in \mathbb{N}$ is not bounded

I'm struggling on how to show that the sequence of $\tan(n)$ is not bounded. Can you please give me some help?
1
vote
2answers
47 views

If $x$, $|x+1|$, $|x-1|$ are three terms of an arithmetic progression, what is its sum upto 20 terms?

I can't figure out how to work with the modulus in this problem. The answer is (180 or 350).
1
vote
1answer
54 views

Does this double series converges?

Let $p>1$ and $q>1$ be two positive integers. Does this double series converges $$\sum_{m=1}^{\infty}\sum_{j=1}^{\infty}\frac{1}{2^{mp+jq}}$$
0
votes
1answer
127 views

How to show all the terms in a sequence are greater than a number?

Prove that if $\left\{ x_n \right\}_{n = 1}^{\infty}$ converges to $3$, then there exists an $N$ such that $x_n > 2$ for all $n \geq N$. I was thinking there are two cases; one where the ...
1
vote
2answers
56 views

Is this a geometric series? If so please help point me in the right direction for calculating the sum: $\sum_{n=1}^\infty n[{1\over 4}]^{n-1}$

Is this a geometric series? If so please help point me in the right direction for calculating the sum: $$\sum_{n=1}^\infty n[{1\over 4}]^{n-1}$$ I know using the test for divergence that this does ...
0
votes
1answer
178 views

I have a question about integrating, and what to do about the constant. $\displaystyle\int\frac{1}{1-z}dz$

http://www.maa.org/sites/default/files/pdf/upload_library/2/Kalman-2013.pdf On page 44, they conclude that $g'(z) = - \displaystyle\frac{\ln(1-z)}{z}$ by saying that it is just ...
1
vote
2answers
101 views

Determine whether the series $\sum\limits_{n=1}^\infty \frac{n!+n}{(n+2)!}$ is convergent or divergent.

Determine whether the series $$\sum\limits_{n=1}^\infty \frac{n!+n}{(n+2)!}$$ is convergent or divergent. Wolfram Alpha says that "By the comparison test, the series converges" but I can't ...
8
votes
3answers
622 views

Find value of sum of reciprocals of powers of a number

Is there a simple way to find the value of the following expression? $$\frac1x+\frac1{x^2}+\frac1{x^3}+\cdots$$ On trial and error, I was getting $\frac1{x-1}$, but I'm looking for a mathematical ...
1
vote
3answers
87 views

Show $\sqrt[n]{b_n}\to 1$, given $\sqrt[n]{a_n} \to 1$ and $\frac{b_n}{a_n} \to g\in (0, \infty)$

Suppose that $\lim_{n\rightarrow\infty}\sqrt[n]{a_n}=1$ and $\lim_{n\rightarrow\infty}\frac{b_n}{a_n}=g$ where $g\in (0,\infty)$. We must prove that $\lim_{n\rightarrow\infty} \sqrt[n]{b_n}=1$. ...
2
votes
2answers
60 views

On $n \times n$ grids filled with $1$ and $-1$

The following question was asked in a contest which I have difficulty proving . Let $n$ be an odd positive integer and suppose that each square of an $n \times n$ grid is filled with either $1$ or ...
1
vote
0answers
44 views

Notation confusion about sum of $\Lambda (n)$

This is hopefully a small point of notation I am missing. I am used to the first two equalities below. $$\sum_{n \geq 1} \Lambda(n) n^{-s} = \sum_{p \mbox{ prime}} \sum_{m \geq 1} \Lambda(p^m) ...
0
votes
2answers
46 views

Calculate $a_n$ with formal power series

I have $$A(x) = \sum_{n=0}^\infty a_n x^n$$ and $$A(x) = (1+x)/(1+7x+6x^2)$$ I need to find $a_0,a_1,a_2,a_3$. I multiply on each side and I get $$ (1+7x+6x^2) (a_0+a_1x+a_2x^2+a_3x^3+\ldots)=1+x ...
0
votes
2answers
42 views

For what values of x does $\sum_{n=1}^\infty {kn \choose ln}x^n$ converge?

I'm having some hard time answering the following question: For what values of $x\in \Bbb{R}$ does this series converge $\sum_{n=1}^\infty {kn \choose ln}x^n$ where $k>l$. Thanks in advance!
1
vote
1answer
68 views

Set theory - Can someone explain sequence operator?

I'm reading up on set theory and relation and I need help understanding this: Two sequences of the same element type can be composed to form a single sequence in such a way that the order of each ...
0
votes
4answers
106 views

Determine the convergence of infinite series $1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-\frac{1}{3} +\cdots$

How shall I determine whether the series $$1-1+ \left(\frac{1}{2}\right)-\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)-\left(\frac{1}{3}\right)+\cdots$$ is convergent or divergent? Please help?
3
votes
1answer
122 views

Is the term “telescoping product” well known?

I know that "telescoping series" (or sum) is well known. But I can't find many reliable references to the term "telescoping product". It would be one of the following: $x_i = \dfrac{y_i}{y_{i+1}}$: ...
1
vote
1answer
84 views

Closed-form expression for a sum of reciprocals of factorials [closed]

Is there a closed-form expression for the finite sum $$\sum_{s=1}^{2^{n-1}}\frac1{(s-1)!}$$ as a function of $n$?
0
votes
0answers
20 views

Consider $f_n(x)=\frac{1}{n^2}I_{[0,n]}(x).$ Discuss the following relations…

Let $I_A(x)$ be the function that is 1 if $x$ is in $A$ and 0 elsewhere. \ Let $f_n(x)=\frac{1}{n^2}I_{[0,n]}(x).$ {problem} \item a) For each $x,$ find $\ds{\lim_{n\to\infty}}f_n(x).$ Define ...
3
votes
2answers
138 views

Deciding whether series containing $a_n$ are convergent knowing that $\lim_{n\rightarrow\infty}\frac{\frac{(-1)^n}{\sqrt{n}}}{a_n}=1$

We know the following thing about sequence ${a_n}$: $$\lim_{n\rightarrow\infty}\frac{\frac{(-1)^n}{\sqrt{n}}}{a_n}=1$$ And now the problem asks us whether it's true for every such $a_n$ that: ...
3
votes
0answers
74 views

Finding the length of an elliptical spiral

Okay, i had a very strange thought, it was "Is it possible to find the length of an elliptical spiral whose major and minor axes were decreasing?" Like for example lets say that $$ \frac{a}{b} = n ...
0
votes
1answer
36 views

Understanding of formula for arithmetic progression

A arithmetic progression is defined by: $$\begin{cases} a_1=a_1 \\ a_n= a_{n-1}+d \end{cases} $$ I can't understand this passage from a algebra book about arithmetic progression. The formula is ...
1
vote
5answers
99 views

Is there a sequence $a_n$ for which $\lim_{n\rightarrow\infty} |\frac{a_{n+1}}{a_n}|$ doesn't exist, but $\sum a_n$ is convergent?

The question is in the title: Is there a sequence $a_n$ for which: $\lim_{n\rightarrow\infty} |\frac{a_{n+1}}{a_n}|$ doesn't exist (so it's not defined, which means we're looking for a case when ...
0
votes
1answer
21 views

Is my algerbra correct for my work dealing with the alternating series test for: $a_n = {(-4)^n \over n4^n} $

I have two questions (see "(1)/(2) Is this valid" sections) below. Given the series: $a_n = {(-4)^n \over n4^n} $ I'd like to see if this converges and think this is an alternating harmonic series. ...
4
votes
3answers
165 views

Closed form of $\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$

How would you recommend me to tackle the series $$\sum_{n=1}^{\infty} \frac{1}{2^n(1+n^2)}$$? Can we possibly express it in terms of known constants? What do you think about it?
0
votes
2answers
32 views

Convergence of sequences in $\mathbb{C}$

I am currently getting into the field of complex numbers, with the imaginary unit $i^2 = -1$ and stuff. At the moment i am looking onto a few sequences in $\mathbb{C}$, regarding convergence. I have ...
0
votes
0answers
56 views

Closed Form for a Sequence

I have come across this sequence $$a_0 = -2, a_1 = 5, a_2 = -28, a_3 = 255$$ and, in general $$a_n = -\frac{1}{2}\bigg(\sum_{i=1}^n \binom{2n+4}{2i}a_{n-i} + \binom{2n+4}{2n+1}\bigg)$$ I've tried ...
1
vote
3answers
59 views

Find the sum of the series with unordered powers of $3$

Consider the following series: $$\sum_{n=1}^{\infty} a_n = 1/3+1+1/3^3+1/3^2+1/3^5+1/3^4+1/3^7+1/3^6 +\dots$$ Determine if it converges, and find the sum. Here is what I got: a) ...
0
votes
1answer
44 views

Parametrized series - how to tackle this problem?

We have the following series: $$\sum_{n=1}^{\infty}\frac{y^n+1}{5^n+n^5}$$ And the problem asks us to say for every $y\in\Bbb{R}$ whether the series is convergent or not (that is - we need to say ...
0
votes
1answer
44 views

Series convergence of $\sum_1^\infty\frac{(n!)^2+(2n)^n}{n^{2n}}$

$$\sum_1^\infty\frac{(n!)^2+(2n)^n}{n^{2n}}$$ Is there a way to test convergence of this series wihout splitting it ? (in case splitting is correct)
0
votes
1answer
30 views

Finding continuous functions from a set

Let $A=\{0,1,\frac{1}{2},\frac{1}{3},...\}$. I want to find continuous functions from $f:A\to \mathbb R$. I proceed in this way. Any sequence converges to $x(\neq 0)$ will be eventually constant ...
0
votes
2answers
400 views

Technique for finding the nth term

I'm a trainee teacher and I need to pass a skills test in maths, but I'm having trouble in this particular area of algebra in finding the nth term of any sequence. I've been given a series of ...
3
votes
2answers
129 views

$a_{n+1}=1+\frac{1}{a_n}.$ Find the limit.

Let $a_1=1$ and $a_{n+1}=1+\frac{1}{a_n}.$ Is $a_n$ convergent? How could i find its limit? I found even terms of the sequence decrease and odd terms are increase. But i cant find upper and lower ...
1
vote
7answers
3k views

Why does the sum of the reciprocals of factorials converge to $e$?

I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with ...
1
vote
3answers
188 views

limit is also bounded if sequence is bounded and converge?

assume $\lim _{n\to \infty }\left(a_n\right)=\:L$,and for every n, $a_n\in \left[c,d\right]$. It is true to say that $L\in \left[c,d\right]$? I can't think of opposite example for this so i think its ...
1
vote
1answer
59 views

Evaluating $\displaystyle\lim_{n \to +\infty}\frac{\left(1 + \sin\alpha\right)^n}{\left(1+\frac{\sin\alpha}n\right)^n}$

Let, for $n \ge 2$, $$a_n = \frac{\left(1 + \sin\alpha\right)^n}{\left(1+\frac{\sin\alpha}n\right)^n}$$ Evaluate $\displaystyle\lim_{n \to +\infty}a_n$, as $\alpha$ varies in $[0, 2\pi)$. I ...
0
votes
1answer
36 views

sum over arbitrary subsets

Let $\{a_n\}_{n\in\mathbb{Z}}$ be a sequence of complex numbers. We want to show that $|\sum_{n\in\mathbb{Z}} a_n|$ (or $\sum_{n\in\mathbb{Z}} |a_n|$) is finite. Is it sufficient to show the ...