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

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2
votes
2answers
39 views

can someone prove my guess from a simple question?

i found an interesting conclusion. when i do this simple question. $f(x)=(x-1)^2(x+2)=x^3+2x^2-x-2$ if $x_i,(i=1,2,3)$ are the roots of $f(x)=0$, solve $\displaystyle \sum_{i=1}^3x_i^2$ ...
3
votes
4answers
82 views

Find $\sum\limits_{n=1}^{\infty}\frac{n^4}{4^n}$

So, yes, I could not do anything except observing that in the denominators, there is a geometric progression and in the numerator, $1^4+2^4+3^4+\cdots$. Edit: I don't want the proof of it for ...
0
votes
0answers
8 views

Help in estimating error in alternating series. (homework)

I tried to do it (4 times already actually) I read that to get the error (upper bound) I should get the value of a(n+1) which in this problem is the value of the term at n=23. But I do not know why am ...
12
votes
3answers
878 views

Infinite Series: Fibonacci/ $2^n$

I presented the following problem to some of my students recently (from Senior Mathematical Challenge- edited by Gardiner) In the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, 34, 55,\ldots$ each ...
1
vote
3answers
47 views

What is solution to this maths series problem?

I found this question on facebook and me and my friend were discussing the possible solution for 9. We have found 3 answers and none of us has any idea which one is correct as all of them looks ...
1
vote
0answers
34 views

The sum of the infinite series [duplicate]

The sum of the infinite series: $$ \frac{1}{2} +\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +\cdots$$ I am able to find ...
2
votes
0answers
16 views

Planar Graph Isomorphism

In 1980, I. S. Filotti & Jack N. Mayer proved planar graph isomorphism testing could be done in polynomial time. Does anyone have an implementation of that? I have a few billion planar graphs ...
28
votes
1answer
375 views

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

There is a known asymptotic expansion of harmonic numbers $H_n$ for $n\to\infty$: $$\begin{align}H_n&=\gamma+\ln n+\sum_{k=1}^\infty\left(-\frac{B_k}{k\cdot n^k}\right)\\ &=\gamma+\ln ...
0
votes
1answer
14 views

Find a geometric progression with sum $100$

I have to find an infinite geometric progression having sum $100$, then to find its first term by assuming that the common ratio is $\frac{1}{4}$. Any hints?
0
votes
1answer
33 views

Series summation of Geometric-Harmonic series

I am trying to find the series summation for the following series : $ \sum_{i=0}^{k} \frac{\beta^i}{i+1}$ and $ \sum_{i=0}^{k} \frac{\beta^i}{(i+1)^2}$ $\beta \in (0,1)$ Any ideas on how to ...
4
votes
8answers
2k views

1, 5, 9, 13, 17, 21,…

How would you describe the set $\{1, 5, 9, 13, 17, 21,\dotsc\}$ in the style of $x:P(x)=$? I know that the sequence is "the last number + 4" or $4n-3$.
0
votes
0answers
30 views

prove this sequence to decreasing for all $n$

Define $a_{n}=1$,and such $$a_{n+1}=a_{n}+\ln{\left(\dfrac{n}{n+1}\right)}+\dfrac{1}{e^{a_{n}}\cdot n}$$ show that $$a_{n+1}<a_{n}$$ or ...
12
votes
4answers
581 views

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. ...
0
votes
1answer
34 views

Vector Space that is a Sequence Space

Consider the sequence $(x_n)_{n \in \Bbb N}$ $ |$ $ x_n=\sum_{i=0}^{n}f(i)$ then if $f(i)$ can also be viewed as a sequence $(y_n)_{n \in \Bbb N}$ $ |$ $y_n=f(n)$. Both of these sequences ...
0
votes
1answer
57 views

Generating function for any series

Given a summation series, is there any way to generate a function to find the value of the sum of first n terms? For example, we have, $\sum f(n) = f(0) + f(1) + ... + f(n)$ . Now, I want to know ...
9
votes
3answers
180 views

If $u_{n+1}\le u_n+u_n^2$ and $\sum u_n$ converges, prove that $\lim\limits_{n\to +\infty}(n\cdot u_n)=0$

Given the positive sequence $\{u_n\},n\in \mathbb{N}$ that meets the conditions: $\boxed{1}$. $u_{n+1}\le u_n+u_n^2$ $\boxed{2}$. Exist the constant $\text{M} >0$ so that ...
11
votes
3answers
706 views

Infinite series $n^7/(\exp(2\pi n)-1)$

I found an interesting topic on this site with regards to the series I am trying to evaluate: Summing $\frac{1}{e^{2\pi}-1} + \frac{2}{e^{4\pi}-1} + \frac{3}{e^{6\pi}-1} + \cdots \text{ad inf}$ I ...
0
votes
0answers
12 views

uniformly convergent sequence of differentiable functions, series of derivative of terms not convergent

I am attempting to come up with a uniformly convergent sequence of differentiable functions $g_{n}:(0,1)\to \mathbb R$ such that the sequence $\{g_{n}'\}$ does not converge. I was thinking that ...
18
votes
7answers
5k views

Are the integers closed under addition… really?

Okay so I'm a 3rd year undergraduate studying Mathematics. I've proved in group theory countless times that the integers are closed under addition. It's obvious to me that they are. However this has ...
3
votes
3answers
196 views

What is the value of $ a_{2009}$

I have the following sequence : $a_0 = 3 $ $ a_n = 2 + a_0 a_1 a_2\text{ ...}a_{n-1} $ How can I find the value of $a_{2009} $ ?
3
votes
3answers
9k views

Proving formula for product of first n odd numbers

I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$): $$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$ How ...
-1
votes
1answer
38 views

What will I pay in month x if I pay 1/36 of balance each month? [on hold]

I have an open note at a bank that I pay 1/36th of the balance every month. I am looking for an equation that will allow me to know what my payment will be on x month.
0
votes
0answers
29 views

Proving “if” direction of continuous iff sequence x_n converging to x implies f(x_n) converges to f(x)

Here is the theorem in mathjax: A real value function $f$ is continuous at $x \in R$ iff whenever a sequence of real numbers $x_{n}$ converges to $x$ then the sequence $f(x_{n})$ $\rightarrow f(x)$. ...
4
votes
0answers
61 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
2
votes
4answers
87 views

Proving that all terms in sequence are positive integers (Recursive)

I am preparing for Putnam and working through practice-like questions and I am boggled. I've got the sequence defined by $a_1 = 1$ and $a_{n+1}$ = $2a_n+\sqrt{3a_n^2-2}$ for any $n\in\mathbb{N}$ and ...
1
vote
3answers
100 views

Find the series: $\frac{-1}{4}+\left(\frac12+\frac14+\frac28+\frac3{16}+\frac5{32}+\cdots\right)$

Find the series: $$\frac{-1}{4}+\left(\frac12+\frac14+\frac28+\frac3{16}+\frac5{32}+\cdots\right)$$ Evidently, this is a Fibonacci Sequence with a Geometric Sequence. But I don't think there is a ...
1
vote
2answers
68 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
0
votes
0answers
15 views

$f:[a,b] \to [0, \infty)$ continuous , then $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ? [duplicate]

Let $f:[a,b] \to [0, \infty)$ be continuous , then is it true that $\lim_{n \to \infty} \Bigg(\int_a^b \big(f(x)\big)^ndx \Bigg)^{1/n}=\sup \{f(x):x \in [a,b]\}$ ?
2
votes
1answer
34 views

How to calculate the closed form of the Euler Sums

We know that the closed form of the series $$\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}\left( {\begin{array}{*{20}{c}} {2n} \\ n \\ \end{array}} \right)}}} = \frac{1}{3}\zeta \left( 2 ...
-1
votes
2answers
71 views

Taylor series expansion, $f(x)=\frac{1}{4-x^2}$

Find taylor series expansion of the following function: $f(x)=\frac{1}{4-x^2}$ In the neighbourhood of the point $x_0=1$ determine the radius of convergence of this series. I do not understand the ...
333
votes
27answers
34k views

Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
0
votes
2answers
55 views

To determine the existence and the value of $\lim_{x \to 0} \frac {2^x-1} x$

I would like to somehow firstly show that $$\lim_{x \to 0} \frac {2^x-1} x$$ exists and determine the value of the limit. My first ideas were by Monotone Convergence. I have been able to prove that ...
1
vote
1answer
23 views

Convergence Proof Help?

Question: Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence $(x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n)$. Prove that $(z_n)$ is convergent if and only if $(x_n)$ ...
1
vote
1answer
12 views

Real analysis: Characteristic property for unconditional divergence

A convergent series $\sum_{k=1}^\infty a_k$ is called unconditional convergent, when it's value is invariant under any permutation $\sigma:\mathbb N\to\mathbb N$ of it's summands, i.e. ...
2
votes
6answers
134 views

Find the limit of the sequence $(4^n)/((2n)!)$

How to find $$\displaystyle \lim_{n\to\infty} \frac{4^n}{(2n)!}$$ Can I use l'Hopital's rule?
0
votes
1answer
29 views

proving convergence of this sequence and calculating limit

So i have this sequence: $a_{n}=1+ \frac{1}{3}\cos1 +\frac{1}{3^2}\cos2+ ... +\frac{1}{3^n}\cos(n) $ I have to prove it is convergent, and then calculate the limit. I'm not totally sure how to find ...
3
votes
1answer
63 views

Prove a sequentially compact metric space is bounded.

Prove that if the metric space $(X, d)$ is sequentially compact, that there exists points $x_0$ and $y_0$ belonging to $X$ such that; $$d(x, y) \leq d(x_0, y_0)$$ for every $x$ and $y$ belonging to ...
-1
votes
1answer
84 views

What is the next number of the following sequence 27, 54, 81, 135, 189,…

What is the next number of the following sequence 27, 54, 81, 135, 189,........ Options Given: 1) 108 2) 243 3) 405 4) 216 5) 378 6) 486 7) 297 8) 459 9) 351 10)None of these My Approach: ...
2
votes
0answers
19 views

The remainder estimate for the integral test

The remainder estimate for the integral test states that if $a_k=f(k)$ where $f$ is a continuous, positive, and decreasing function on $[n,\infty)$ and $R_n=s-s_n$ (where $s_n$ is the $n$th partial ...
1
vote
1answer
27 views

What is the Limit of the following Fibonacci Sequence?

The Fibonacci numbers $x_1,x_2,.......,$ are defined recursively by $x_1=1, x_2=2$ and $x_{n+1}=x_n+x_{n-1}$ for $n\geq2$. Show that, $\lim_{n \to \infty}\frac{x_{n+1}}{x_n}$ exists, and evaluate the ...
25
votes
9answers
3k views

How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2?

Prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2 My attempt I proved that the sequence is increasing and bounded by 2, can anyone help me show that the ...
2
votes
1answer
87 views

Fourier-Bessel series of $f(x)=x^2$

I'm trying to calculate the expansion of $f : [0,1]\to\mathbb{R}$ given by $f(x)=x^2$ in a Fourier-Bessel series of zeroth order. In that case let $J_0$ be the $0$-th order Bessel function and ...
0
votes
1answer
67 views

what's the limit $\lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2}$

I have no idea how to compute the tail sum $\lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2} $. I tried subtracting the first n items from all but realized that I don't know a way to calculate ...
0
votes
1answer
21 views

Proving convergence of a sequence through a polynomial

Let ($x_n$)$\rightarrow$x and let $p(x)$ be a polynomial. (a) Show $p(x_n)$$\rightarrow$$p(x)$. (b) Find an example of a function $f(x)$ and a convergent sequence ($x_n$)$\rightarrow$x where the ...
1
vote
0answers
55 views

Analytic representation of Harmonic numbers

As we know, using $$\frac{{{{\ln }^2}\left( {1 - x} \right)}}{{1 - x}} = \sum\limits_{n = 1}^\infty {\left( {H_n^2 - {\zeta _n}\left( 2 \right)} \right){x^n}} = \sum\limits_{n = 1}^\infty {\left( ...
3
votes
1answer
62 views

Can the Fibonacci sequence be written as an explicit rule?

When I learned sums and sequences in algebra II with trig I learned about recursive rules and explicit rules. A recursive rule written with the formula of: $$a_n = r * a_{n-1}$$ Or as: $$a_n = ...
1
vote
2answers
59 views

How to solve $\sum_{k=1}^n\frac{k}{n^2+k}$?

Can someone show me what is wrong with the expression I got for evaluating $\sum_{k=1}^n\frac{k}{n^2+k}$? Steps: $\sum_{k=1}^n\frac{k}{n^2+k} = \frac{\sum_{k=1}^nk}{\sum_{k=1}^{n}n^2+k} = ...
2
votes
0answers
39 views
+50

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
1
vote
2answers
148 views

Solve this integral:$\int_0^\infty\dfrac{\arctan x}{x(x^2+1)}\mathrm dx$

I occasionally found that $\displaystyle\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\dfrac{\pi}{2}\ln 2$. I tried that $$\int_0^{\frac{\pi}{2}}\dfrac{x}{\tan x}=\int_0^{\frac{\pi}{2}}x \ \mathrm ...
0
votes
3answers
47 views

Showing that a sequence is unbounded

How do I show this sequence is unbounded. ${b_j=j}$ from j=1 to infinity By using the following definition. ${b_j}$ is called bounded if there exist $M>0$ such that $b_j<M$ for all $j\in$ ...