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

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0
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2answers
163 views

Limit of $(1+5/n+6/n^2)^n$ when $n$ goes to infinity [closed]

Find $$\lim_{n \to \infty} \left(1+\frac{5}{n}+\frac{6}{n^2}\right)^n$$
0
votes
1answer
20 views

Is that possible for an array to be divergent sequence but convergent series?

Is there possible to construct an array such that when it is consider as a sequence, it diverges. But as series, it converges ??
1
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1answer
50 views

Proof of divergence of a series

I'd really appreciate some help with this question on my recent math assignment: Show that if $a_n > 0$ and $\lim_{n\to \infty} na_n = L$, where $0 < L < \infty$, then $a_n$ is divergent. ...
0
votes
3answers
68 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
0
votes
2answers
61 views

Sequence and convergence of subsequences

Suppose that $(a_n)$ is a sequence. Assume that both $(a_{2n})$ and $(a_{2n+1})$ converge to the same $L$. Prove carefully that $(a_n)$ also converges to $L$ I was thinking that $(a_{2n})$ and ...
0
votes
1answer
25 views

Inverse theorem on product of two convergent sequences

Suppose I have two sequences, $a_n$ and $b_n$. I know that: $\lim_{n\to\infty} a_n=1$ and that $\lim_{n\to\infty} a_nb_n=c$. Does this mean that $\lim_{n\to\infty} b_n$ converges? If so, by ...
0
votes
1answer
53 views

How to solve recursive function

I've recently been doing some limits with circuits and such, and I came up with the following equation, $R$ being a constant: $$f(x) = \frac{f(x-1)*R}{R+f(x-1)}+R$$ with $f(1)=2$. I know that this ...
1
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1answer
44 views

How to construct longitudinal from transversal waves and vice versa?

The above construction of a longitudinal wave out of a transversal wave has been encountered somewhere in an old physics textbook. There are several drawbacks with this construction. The maximum ...
0
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1answer
156 views

The meaning of almost surely convergence

Consider the space of sequences of tosses of a fair coin. In particular let $X_{nH}$ be the number of times that a coin lands on heads in a sequence of $n$ coin flips. Consider statement $S$ below. ...
0
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0answers
64 views

Help for applying Hahn Banach theorem in this exercise .

I want to use this version of Hahn Banach: $X$ real vector space, $M$ a subspace, $p$ a sub-linear mapping and $T:M\rightarrow\mathbb{R}$ linear and $T(x)\leq p(x)$ $\forall x \in M$ Then ...
1
vote
3answers
49 views

problem with pattern of sequance

there is a sequence $a_1=1^2, a_2=3^2, a_3=6^2, ...$ I'm thinking if there is a pattern of this progression, but so far I haven't find out. I noticed that the difference of only the square number ...
0
votes
1answer
39 views

how prove there are infinite numbers $\frac{a_{i}}{1234}\in N$ and $a_{n+2}=a^2_{n+1}-a_{n}$

let $a_{1}=287,a_{2}=39$,and $$a_{n+2}=a^2_{n+1}-a_{n}$$ show that: this sequence $\{a_{n}\}$ contains infinitely many $a_{i}$,such that $\dfrac{a_{i}}{1234}\in N$ My try: since ...
0
votes
2answers
271 views

Show $a_{n+1}=\sqrt{2+a_n},a_1=\sqrt2$ is monotone increasing and bounded by $2$; what is the limit?

Let the sequence $\{a_n\}$ be defined as $a_1 = \sqrt 2$ and $a_{n+1} = \sqrt {2+a_n}$. Show that $a_n \le$ 2 for all $n$, $a_n$ is monotone increasing, and find the limit of $a_n$. I've ...
2
votes
2answers
107 views

Uniform (but not normal) convergence of a series of function

consider the series of function $\sum f_n$ with $f_n(x)=\frac{x}{x^2+n^2}$. It is easy to see that there is pointwise convergence on $\mathbb{R}$ (to a function that we'll call $f$) but not normal ...
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votes
1answer
70 views

If the series $\sum_{k=0}^{\infty} a_k$ converges then $a_k$ converges to 0. [duplicate]

If the series $\sum_{k=0}^{\infty} a_k$ converges then $a_k$ converges to 0. How to prove this theorem?
5
votes
3answers
321 views

Can any continuous function be represented as an infinite polynomial?

Can any continuous function be represented as an infinite polynomial? Motivation: the antiderivative $ \int^\ e^{-x^2}dx\ $ can be expressed as an infinite polynomial(write Taylor series for ...
-1
votes
5answers
103 views

Which sequence will be longer? sequence of natural numbers or even numbers [closed]

Its not that I don't know answer for this question, but this question is definitely debatable. the main reason for asking this question is that someone will be able to convince me on the answer as I ...
0
votes
1answer
39 views

Show that: $\sum_{n=1}^{\infty} n(n-1)s^{n-2} = \frac{2}{(1-s)^3}$

How can I show that: $\sum_{n=1}^{\infty} n(n-1)s^{n-2} = \frac{2}{(1-s)^3}$ I'm struggling to figure out how to start on this question. Should I sum the series and then differentiate it and also ...
2
votes
1answer
54 views

Finding the value of $a^x$

We have the series expansion $e^x = 1 + \frac{x}{1!}+ \frac{x^2}{2!}+ \frac{x^3}{3!}+ \frac{x^4}{4!}...\infty$. Is it possible to write $a^x$ in the similar form, where ...
5
votes
2answers
175 views

Infinite series $\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$

How can I find the value of the following sum? $$\sum_{n=0}^{\infty}\arctan(\frac{1}{F_{2n+1}})$$ $F_n$ is the Fibonacci number.($F_1=F_2=1$)
5
votes
2answers
864 views

Series $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$

Is it true that for $x\in[0,2\pi]$ we have $$\sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}=\frac{x^2}{4}-\frac{\pi x}{2}+\frac{\pi^2}{6}$$ How can I prove it? For other intervals what is the value of above ...
1
vote
1answer
78 views

A problem on sequence and series

Suppose we have a decreasing sequence $\{x_n\}$ which converges to $0$. Then is it true that the sum $$\sum_{n=1}^{\infty}\frac{x_n-x_{n-1}}{x_n}$$ diverges ?
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2answers
83 views

Evaluation of a series (possibly related to Binomial Theorem)

I have the following series: $$1 + \frac{2}{3}\cdot\frac{1}{2} + \frac{2\cdot5}{3\cdot6}\cdot\frac{1}{2^2} + \frac{2\cdot5\cdot8}{3\cdot6\cdot9}\cdot\frac{1}{2^3} + \ldots$$ I have to find the ...
5
votes
4answers
206 views

Why are these equations equal?

I have racked my brain to death trying to understand how these two equations are equal: $$\frac{1}{1-q} = 1 + q + q^2 + q^3 + \cdots$$ as found in ...
0
votes
2answers
66 views

Determine the value for which a sequence is an arithmetic progression.

We have the following sequence $$ -a, -\dfrac{a}{b}, \dfrac{a}{b}, a$$ Determine the value of $b$ for which this is an arithmetic progression ($a \neq 0$) I don't know how to do this. I've tried ...
1
vote
3answers
39 views

Find the series expansion of 2 multiplied functions

The first three terms in the series expansion of $(1+x)^m$ are $1 + mx + \dfrac{m(m-1)x^2}{2}$. Find the first 3 terms in the series expansion of $(1+x)^{m+1}(1-2x)^m$. I don't really know ...
2
votes
1answer
67 views

The sequence $H_n-\ln(n)$ converges

Is there a proof that the sequence $\displaystyle \sum_{k=1}^n \frac{1}{k}-\ln(n)$ converges that doesn't use integrals?
1
vote
2answers
71 views

Convergent Sequence Counterexample

$\lim_{n\to\infty}s_n = 0$ iff $\lim_{n\to\infty}|s_n| = 0$. I know this statement is false but I need to find a counterexample and I can't think of one. Any hints??
0
votes
5answers
100 views

Proof of the limit of a sequence

The sequence is: $a_n$ = $\frac 1n$ [$(\frac 1n)^2 + (\frac 2n)^2 + (\frac 3n)^2...(\frac nn)^2$] The objective is a proof of the limit from 1 to infinity. Just from toying around with a few ...
3
votes
5answers
106 views

Show that $\sum_{i=0}^{n-1} 2^i = 2^n - 1$

This is a problem from a book with no solution. Show that: $$\sum_{i=0}^{n-1} 2^i = 2^n - 1$$ without using the formula for geometric series. My lengthy solution is as follows: Try with $n = 1, 2, ...
2
votes
0answers
92 views

Is there a “natural” norm in the space of all sequences?

Let $X$ be $\mathbb{R}^\mathbb{N}$, the space of all real-valued sequences. Is there a natural norm in this space?
1
vote
1answer
153 views

Concrete Mathematics - Stability of definitions in the repertoire method

There are some existing questions on the repertoire method from the first chapter but I think I'm stuck on something different than the part people usually have trouble with. I think the jump in the ...
1
vote
0answers
108 views

Maximum of a sequence of almost-identical independent normal random variables

Take a sequence $X_1,\ldots,X_n$ where each $X_i\sim\mathcal{N}(\mu,\sigma^2)$ is an i.i.d. normal random variable. Denote by $X_\max$ the maximum of this sequence. A well-known fact about ...
1
vote
9answers
187 views

Finding a formula for a repeating sequence of 1's and -1's

Is there a simple formula for the sequence $(a_n)$ given by $(1,1,-1,1,1,-1,1,1,-1,\cdots)$ (with the repeating pattern 1,1,-1), starting with $n=1$?
0
votes
0answers
31 views

Limits in a topological space

Consider the topological space $(\Bbb R^2, \tau)$ described by the following neighbourhood base: $\beta_{(x,y)}= \{(x-\epsilon,x+\epsilon) \times (y-\epsilon, y+\epsilon)\subseteq \Bbb R^2: ...
10
votes
1answer
209 views

Limit of maximum of $f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)})$

let $$f_{n}(x)=\dfrac{1}{n}(\sin{x}+\sin{(2x)}+\sin{(3x)}+\cdots+\sin{(nx)}),x\in R,n\in N$$ let $$a_{n}=\max_{x\in R}{(f_{n}(x))}$$ Find this limit $$\lim_{n\to\infty}a_{n}$$ My try: since ...
0
votes
1answer
80 views

Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as a(n) = a(n-1) - 1/a(n-1), where a(0) is given. ...you already know what I'm asking you ...
0
votes
1answer
62 views

How do I come to a series expansion of $1/(e^z-1)^2?$

How do I come to a series expansion of $\frac{1}{(e^z-1)^2}?$ $e^z-1$ can be expanded to: $$1 + z + \frac{1}{2} z^2 + \frac{1}{6} z^3 + \dots -1$$ so the series becomes: $$\frac{1}{(z^2 (1 + ...
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votes
1answer
245 views

Convergence of complex sequences

Do the following sequences converge? 1) $(-1)^n\frac{n}{n+i}$ 2) $\frac{n^2+in}{n^2+i}$ I don't really understand how to decide whether a complex sequence converges, and I don't have much ...
2
votes
1answer
37 views

Increasing sequence by derivative?

I prove that f is an increasing function like that: $\exists d. (0 < d) \wedge \forall p. (0 < p \wedge p < d) \Rightarrow (f(x) < f (x + p))$ Now, I would like to prove that the ...
7
votes
2answers
75 views

How prove this limit $\lim_{n\to\infty}\frac{n^2}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}}=0$

Assume that a postive term series $\displaystyle\sum_{n=1}^{\infty}a_{n}$ coverges, show that $$\lim_{n\to\infty}\dfrac{n^2}{\dfrac{1}{a_{1}}+\dfrac{1}{a_{2}}+\cdots+\dfrac{1}{a_{n}}}=0$$ My try: ...
2
votes
1answer
196 views

How do I use telescopic cancellation to prove the Fibonacci Sum Identity

I am reading a textbook which attempts to prove the Fibonacci Sum Identity by rearranging the Fibonacci recurrence relation as follows and then using telescoping cancellation to prove the identity: ...
1
vote
1answer
92 views

Proving Convergent Series made by continuous function $f$, $a_n=f(1/n)$

Let $f$ is continuous on an interval around 0, and let $a_n=f(1/n)$ Prove that if $f''(0)$ exists and $f(0)=f'(0)=0$, then $\displaystyle{\sum_{n=1}^{\infty}{a_n}}$ converges. This is problem from ...
1
vote
1answer
193 views

Prove that a sequence diverges if and only if its subsequence diverges

Prove that $(x_n)_n$ diverges if and only if for every $a\in\mathbb{R}$, there exists an $\epsilon > 0$ and a subsequence $(x_{n_k})_k$ for $(x_n)_n$ such that for all $k\in \mathbb{N}$, $|{x_{n_k} ...
1
vote
6answers
63 views

How Do I show a sequence is bounded?

Given the sequence $(S_n)$, such that $S_0 = 1$ and $$S_{n+1} = \frac{S_n}{1 + S_n}$$ show that it is convergent? We were able to show that it was monotone but we are not sure how to show that it ...
1
vote
1answer
53 views

Proving the form of a sequence's terms

How do I go about attacking this problem and what is it asking? Suppose that $\alpha^2 = \alpha + 1$ and suppose $F_n$ denotes the Fibonacci sequence. Show that $\alpha^3 = 2\alpha +1, \alpha^4 = ...
6
votes
3answers
222 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
1
vote
2answers
95 views

write formula to predict nth term of sequence $1, 1\cdot3, 1\cdot3\cdot5, 1\cdot3\cdot5\cdot(2n-1)$

How can I write a formula for a sequence with the following behavior: {$1, 1\cdot3, 1\cdot3\cdot5, 1\cdot3\cdot5\cdot7, 1\cdot3\cdot5\cdot7\cdot9$} 1st term is $1$ 2nd term is $1 \cdot 3 = 3$ 3rd ...
0
votes
1answer
34 views

experimental sequence of number

I'm doing a small numerical experiment. I got, from the first simulations, the following sequence of numbers. I'm trying to imagine a mathematical law behind this sequence. It could be a geometric ...
0
votes
1answer
31 views

Holder Space series term

Holder space $C^{k,\gamma}(\bar{U})$ has the norm $||u||_{C^{k,\gamma}(\bar{U})} := \sum_{|\gamma | \leq k}||D^{\alpha}u||_{C(\bar{U})} + \sum_{|\gamma|=k}[D^{\alpha}u]_{C^{0,\gamma}(\bar{U})}$, where ...