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

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4
votes
4answers
380 views

Geometric series sum $\sum_2^\infty e^{3-2n}$

$$\sum_2^\infty e^{3-2n}$$ The formulas for these things are so ambiguous I really have no clue on how to use them. $$\frac {cr^M}{1-r}$$ $$\frac {1e^2}{1-\frac{1}{e}}$$ Is that a wrong ...
1
vote
3answers
64 views

Does the series converge? If so, find the sum.

$$\sum^\infty_{n=1}\frac{1+2^n}{3^n}$$ I think it can be shown to converge because $r=\frac23<1$. But I can't seem to get the correct sum, which is $\frac53$.
3
votes
2answers
101 views

Equality involving $\sum_n \sin(\gamma_n \log x)/\gamma_n$

This is I think an algebra confusion about an equality of Littlewood, $$\frac{\psi(x) - x}{\sqrt{x}} = -2\sum_{1}^{\infty}\frac{\sin( \gamma_n\log x)}{\gamma_n} + O(1).\hspace{20mm}(1)$$ He refers ...
3
votes
3answers
259 views

Determine convergence or divergence $\sum_1^\infty \sin \frac{1}{n}$ [duplicate]

$$\sum_1^\infty \sin \frac{1}{n}$$ So now I konw that to evaluate this I can just look at the limit as it reaches infinity. I see that it would result in 1 over 0, but it approaches 0 so I could that ...
-5
votes
1answer
134 views

Convergence of $ \sum_{x=0}^\infty\sin(x)$ [closed]

I know that this is a repeating series, but how do I prove that this diverges from 0 to infinity? $$\sum_{x=0}^\infty\sin(x)$$ I really don't understand all this sequences and series stuff, without ...
1
vote
2answers
57 views

Limits of the comparison test

http://en.wikipedia.org/wiki/Direct_comparison_test I don't understand how this works, I understand the idea behind it. Similar to the squeeze theorem it is pretty logical and easy to visually see. ...
0
votes
0answers
43 views

Outer Lebesgue measure - countable sub-additivity

Denote $$ m^*(\Omega) := \inf \left \{ \sum_{k=0}^\infty vol(B_k) : (B_k)_{k=0}^\infty \text{ covers } \Omega \right \} $$ where $\Omega \subseteq \mathbb R^n$ and the $B_k$ are open boxes of the form ...
0
votes
1answer
91 views

Determine the value

The sequence $(u_n)$ is given recursively as follows: $$\begin{cases} u_1=\frac{1}{1+x}\\ \frac{1}{u_{n+1}}=\frac{1}{u^2_{n}}-\frac{1}{u_n}+1, &\forall n\geq 1 \end{cases}$$ where ...
1
vote
0answers
85 views

Converse of Stolz-Cesaro

It is not difficult to prove a converse to Stolz-Cesaro in the form: If $a_n$ and $b_n$ are sequences satisfying: a) $b_n$'s are increasing and divergent b)$\lim_{n\rightarrow ...
0
votes
2answers
62 views

How can we describe different sorts of limits using ordinals, and what kind of “work” can they do in transfinite induction?

I recently encountered transfinite induction and ordinal numbers for the first time in a real analysis text (Bruckner, Bruckner, and Thomson) and I am still coming to grips with these tools and their ...
1
vote
1answer
58 views

Solving $ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) $?

I need to find $F_{n}$ in : $$ F_{n} = \sum_{i=1}^{n-1} (F_{i}\cdot F_{n-i}) , F_0 = 0 , n>=2 $$ This equation screams convolution , I think , but I find it as a quite long solution sometimes. ...
0
votes
2answers
179 views

Question on compound interest

If you deposit $100 at the end of every month into an account that pays 3% interest per year compounded monthly,the amount of interest accumulated after months is given by the sequence. I tried the ...
1
vote
3answers
74 views

Find $F_{n}$ in : $F_{n} +2F_{n-1} + … + (n+1)\cdot F_{0} = 3^{n}$

I'm stuck with the question for a while : Find in $$F_{n} +2F_{n-1} + ... + (n+1)\cdot F_{0} = 3^{n}$$ the element $F_{n}$ . Placing $n-1$ instead on $n$ results in : $$F_{n-1} +2F_{n-2} + ... + ...
1
vote
1answer
65 views

Finding the lowest common value in repeating sequences

Assume I have N sequences of ones and zeros. Each sequence repeats every p terms. I want to find the minimum position where all ...
1
vote
3answers
156 views

How to solve $ \lim_{n\to\infty} \frac {(n!)^\frac{1}{n}}{n} $? [duplicate]

I need to find the limit for: $ \lim_{n\to\infty} \frac {(n!)^\frac{1}{n}}{n} $ I know the answer is $\frac {1}{e}$ but I have no idea how to get that answer. I'd appreciate some help.
0
votes
1answer
96 views

How to solve this arithmetic question?

The sum of 5 consecutive terms of an arithmetic series is 30 and the sum of the squares of these terms is 220. Find the terms.
3
votes
2answers
152 views

Find a formula for this sequence (and prove it).

This is a 2 part problem. Part I I need help finding a formula for this sequence of numbers: $$\frac{1} {1\times 2} + \frac {1} {2\times3} + \cdots + \frac {1} {n(n+1)}$$ Part II I need to prove ...
0
votes
2answers
47 views

Rounding to the nearest term in a geometric progression

Consider the following progression: R(i) = 5*10^(i/30) where i is ith number within the progression. I would like to devise an equation that will round input ...
7
votes
2answers
161 views

An article from AMM vol 95 Page 942. (I have some doubt about it)

Today I am reading an article in American Math Monthly volume 95 Page 942. Author introduces an alternating series:$$\sum_{n=1}^{\infty}(-1)^n\frac{(2n)!}{4^n(n!)^2}$$ then he uses the Stirling's ...
1
vote
2answers
90 views

Holomorphicity of $\zeta(s)$

Let $\zeta(s)=\sum_{n=1}^{\infty}1/{n^s}$ be the Riemann zeta function, $s\in \mathbb{C}$. I can show that it converges absolutely and uniformly on the right half-plane Re$(s)>1$. How can I show ...
2
votes
5answers
1k views

Convergence of a series implies the convergence of the squares

Suppose $a_n\in\Bbb{R}$ and $\sum a_n$ converges. Is it necessarily true that $\sum (a_n)^2$ converges? I think the answer is yes but I was unable to prove it. It would be much easier if $\sum a_n$ ...
1
vote
1answer
171 views

Does the infinite product $\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$ diverge to zero or some other finite value.

Does the infinite product diverge to zero or some other value? $$\prod_{n \mathop = 1}^\infty {\frac{2^n}{3^n}}$$
0
votes
2answers
52 views

Prove an infinite series hypothesis

If I have a hypothesis that this function will work for any given a or b value (greater than zero and one respectively), how would I go about "proving" that... What exactly is considered a "proof" in ...
3
votes
1answer
72 views

How to Find the Radius of Convergence for This Proof?

Prove that if $\sum_{k=0}^{\infty}a_k$ converges, $\sum_{k=0}^{\infty}{a_k}{x^k}$ converges uniformly on $[0, 1]$. I posted this question a few days ago and was given a clue. I think that I'm ...
1
vote
1answer
121 views

How to find number of squares in a chess board

Problem : An $ n\times n$ chess board is a square of side $n$ units which has been sub-divided into $n^2$ unit squares by equally spaced straight lines parallel to the sides. Find the total number ...
1
vote
1answer
66 views

How often can a Riemann rearrangement give the same result?

(From wikipedia) The Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series is ...
1
vote
1answer
172 views

Is the series with positive terms convergent if $na_n \rightarrow 0$?

I am unsure how to tackle this problem: If $\{a_n\}$ is a sequence of positive numbers with $\lim_{n\rightarrow \infty } na_n=0$, what can you conclude about the convergence of $\sum a_n$? (*) First ...
0
votes
2answers
71 views

Convergence of a series containing logarithm

I'm stuck on the following problem: for what values of $\alpha$ the following sum is convergent:$$S(\alpha)=\sum_{k=2}^{+\infty}\frac{1}{\ln(k)^\alpha}$$ If the series is convergent, there will be a ...
1
vote
4answers
91 views

Finding the value to which a sequence converges

The question is $f_1=\sqrt2 \ \ \ , \ \ f_{n+1}=\sqrt{2f_n}$, I have to show that it converges to 2. The book proceeds like this: let $\lim f_n=l$. We have, $f_{n+1}=\sqrt{2f_n} \implies ...
1
vote
0answers
85 views

Prove the convergence of binomial series.

We already have the conclusion that $$(1+x)^{\alpha}=\sum_{n=0}^{\infty}\begin{pmatrix}\alpha\\ n\end{pmatrix}x^n,$$ $x\in(-1,1)$ when $\alpha\leq-1,$ $x\in(-1,1]$ when $-1<\alpha<0,$ ...
0
votes
3answers
72 views

Disproving that a sequence is Cauchy's Sequence

$f_n=(-1)^nn$ I tried doing it this way : Given $\epsilon>0$ we have for $n>m$ $|f_n-f_m|=|(-1)^nn-(-1)^mm|\le |(-1)^nn|+|(-1)^mm|=n+m>2m$ Am i good this far ? Now how do i conclude with ...
2
votes
1answer
170 views

How many flowers do we need?

There are seven temples. We want to give same number of flowers to the each temple. Before giving flowers to any temple, we wash them. Whenever we wash our flowers, the amount of flowers we have is ...
2
votes
2answers
98 views

How to make this difference equation continuous?

$$ F_n=F_{n-1}-F_{n-2} $$ How can I convert this oscillating sequence into a continuous function? IE get it in terms of n.
9
votes
1answer
273 views

Can a sequence of functions have infinity as limit exactly at rationals?

Someone asked me this question. And he said it's an exercise from Rudin's Real and Complex Analysis. Does there exist a sequence of continuous functions $f_n(x)$, such that $\lim_{n \to \infty} ...
1
vote
2answers
964 views

Convergence of series, comparison test on $\frac{1}{\sqrt{n}+ \ln n}$

I am suppose to use the limit comparison test to prove divergence or convergence. There isn't really any examples in my book that show how to pick your $b_n$ so I just pick whatever works out nicely. ...
0
votes
2answers
5k views

Does the series sin(1/n) from 1 to infinity converge? [duplicate]

Does the series $\sum_{i=1}^\infty \sin \frac{1}{n}$ converge? Or does it diverge?
2
votes
2answers
173 views

If $f(t) = 1+ \frac{1}{2} +\frac{1}{3}+…+\frac{1}{t}$, find $\sum^n_{r=1} (2r+1)f(r)$ in terms of $f(n)$

If $f(t) = 1+ \frac{1}{2} +\frac{1}{3}+....+\frac{1}{t}$, Find $x$ and $y$ such that $\sum^n_{r=1} (2r+1)f(r) =xf(r) -y$
2
votes
1answer
54 views

Convergence of $\varphi_n(x):=\frac{\varphi(nx)}{n}$ in Schwartz space

I want to find all $\varphi\in\mathcal S(\mathbb R)$ for which the sequence $\varphi_n(x):=\frac{\varphi(nx)}{n}$ converges in $\mathcal S(\mathbb R)$. The first step, I have already managed to do by ...
0
votes
1answer
911 views

Supremum and Infimum of functions

I have been given the following homework problem... struggling. Any help would be appreciated. With the following functions state; a) State if the function is monotone. 1 b) Decide if it is ...
6
votes
4answers
923 views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
3
votes
1answer
78 views

Deriving the series formula for the digamma function using the functional equation

By repeatedly applying the functional equation $ \displaystyle\psi(z+1) = \frac{1}{z} + \psi(z)$ I get that $$ \psi(z) = \psi(z+n) - \frac{1}{z+n-1} - \ldots - \frac{1}{z+1} - \frac{1}{z}$$ or ...
2
votes
2answers
59 views

The space of sequences of integers, and an analog of topological space for classes?

Is the collection of integer sequences a set or a class? If it's not a set, then is there an analog of topological spaces for classes? Thank you!
0
votes
2answers
84 views

Problem on convergence of sequences

Given that $\lim f_n=1>0$, Show that there exists a positive integer $m$ such that $f_n\ge 0 \\ \forall n \ge m $
1
vote
0answers
65 views

Infinite product of an analytic function on the right half plane

I want to know whether $G_t=\prod_{i=1}^\infty G(s)^i$ is defined and analytic on the right half plane for an analytic $G(s)$ on the right half plane. If so, is $L^{-1}\left(G_t\frac{1}{s}\right)$ ...
1
vote
1answer
155 views

Check the convergence of $ \sum_{n=1}^{\infty} \ln(x)^n$

I`m trying to check the domain of $R$ for $$ \sum_{n=1}^{\infty} \ln(x)^n$$ so what I did is to take $a_n$ and he is $1$ so $\rightarrow -1<x<1$ $$ -1<\ln(x)<1 \longrightarrow ...
2
votes
1answer
177 views

The completeness of the real numbers with respects to Cauchy sequences?

My Question: I am not sure about the very last inequality in the proof below; namely, where did we get $\mid a_{n}-a_{N}\mid$ and $\mid a_{N}-b\mid$? I see that $\mid a_{n}-a_{N}\mid<\epsilon/2$ ...
5
votes
1answer
98 views

Dirichlet vs. logarithmic density

The Dirichlet density of A relative to B is $$ \lim_{s\to 1^+}\frac{\sum\limits_{n\in A}n^{-s}}{\sum\limits_{n\in B}n^{-s}} $$ and the logarithmic density of A relative to B is $$ \delta(A) = ...
-1
votes
5answers
361 views

Determine whether the series converges or diverges [closed]

$$\sum_n \frac{4^{n+1}}{3^n-2}$$ I have trouble when n is used as an exponent, and don't really know how to approach these types of problems. A walk through of how to solve this would be appreciated. ...
3
votes
1answer
88 views

How many numbers can I make with subseries of $\sum_{n=1}^{\infty} \frac{1}{2^n}$?

Given $\sum_{n=1}^{\infty} \frac{1}{2^n}$, what real numbers in $\left[ \frac{1}{2},1 \right]$ can I generate with subseries of this series? Obviously we have every power of $\frac{1}{2^n}$ (by ...
0
votes
1answer
93 views

How many steps will it take to reach the wall?

If a wall is 100 feet away and you step 2 feet, and then cut the length of your step in half each time you step. How many steps would it take to reach the wall?