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

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0
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0answers
29 views

Finding patterns in seemingly arbitrary pairs of numbers

I don't work (directly) in mathematics (I'm a programmer), but I see numbers every day. Today I came across an issue where some totals were off, and was sent a list of the last 9 examples of the ...
0
votes
1answer
35 views

Why does convergence is required for one series to be differentiable?

First of all , I'll let you know that I am really really bad at calculus so please be gentle. Lets have this series: $\sum_{n=0}^\infty \frac{(-1)^{n-1}x^{2n}}{n(2n-1)}$ The thing is I know ...
2
votes
1answer
18 views

Discrete analogue of bounded variation

What kind of sequences $(a_n)\subset\mathbb{R}$ are expressible as the difference of two increasing sequences?
-4
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1answer
67 views

Show that $ H_n \leq H_{\left\lfloor\frac{n}{2}\right\rfloor}+1$

Consider the harmonic sequence $$H_n = 1 + \frac{1}{2} + \frac{1}{3} +\frac{1}{4} + \ldots + \frac{1}{n}$$ I would like to prove that $$ H_n \leq H_{\left\lfloor\frac{n}{2}\right\rfloor}+1.$$
0
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0answers
15 views

Average of Convergent Sequences Proof [duplicate]

Show that if ($x_n$) is a convergent sequence, then the sequence given by the averages: $y_n$=($x_1$+$x_2$+...+$x_n$)/n also converges to the same limit. I know that for all $\epsilon$$>$0, ...
2
votes
1answer
29 views

Floor and Ceiling Series (I) [on hold]

Prove or disprove: 1) $$\sum_{n=2}^{\infty}{\frac{1}{\lfloor n^2/2 \rfloor}} = \frac{1+\zeta(2)}{2}$$ 2) $$\sum_{n=1}^{\infty}{\frac{1}{\lceil\ n^2/2 \rceil}} = \frac{\zeta(2)}{2} + \frac{\pi}{2} ...
2
votes
0answers
23 views

Infinite Bessel function sum

Let $$f(x)=\sum_{p=0}^\infty J_p(x)$$ We have that $f(0)=1$ and a Bessel function like curve. Can $f(x)$ be expressed in terms of known functions? $$f(x)=1-\sum_{n=0}^\infty ...
5
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0answers
23 views

For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
0
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0answers
10 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 ...
5
votes
3answers
64 views

If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$

I found an interesting conclusion when I did this simple question. Let $$f(x)=(x-1)^2(x+2)=x^3+2x^2-x-2$$ and let $x_i$ for $i=1,2,3$ be the roots of $f(x)$. Find the sum $\sum\limits_{i=1}^3x_i^2$. ...
3
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0answers
23 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 ...
1
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0answers
36 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 ...
-5
votes
3answers
81 views

What is solution to this maths series problem? [on hold]

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 ...
4
votes
4answers
97 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
1answer
16 views

Find a geometric progression with sum $100$ [on hold]

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
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0answers
36 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 ...
0
votes
1answer
36 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
0answers
14 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 ...
0
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1answer
61 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 ...
-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
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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
66 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 ...
5
votes
8answers
3k 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 $4n-3$.
9
votes
3answers
190 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 ...
1
vote
3answers
101 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 ...
0
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0answers
16 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
73 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 ...
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. ...
0
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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 ...
-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: ...
1
vote
1answer
28 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 ...
2
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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 ...
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 ...
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 ...
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
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0answers
63 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( ...
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 $(x_n)$ ...
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
63 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} = ...
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 $j\in$ ...
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votes
1answer
18 views

Proving a this sequence converges using epsilon definition

I am not certain how to prove this sequence converge using the epsilon definition Here is the sequence $a_j=\frac{j^2+3}{2j^2-j+9}$ I turned the sequence into a function and for the limit I got $j ...
-5
votes
0answers
22 views

How to study the nature of this series? [on hold]

Can somebody please help me to study the nature of this series? https://goo.gl/YBYSbM
2
votes
3answers
81 views

Evaluate the summation $\sum_{k=1}^{n}{\frac{1}{2k-1}}$

I need to find this sum $$\sum_{k=1}^{n}{\frac{1}{2k-1}}$$ by manipulating the harmonic series. I have been given that $$\sum_{k=1}^{n}{\frac{1}{k}} = \ln(n) + C$$ where $C$ is a constant. I have ...
0
votes
2answers
45 views

Root test for convergence: $\displaystyle{\lim_{n\to\infty} (a+bi)^n}$

$$\lim_{n\to\infty} (a+bi)^n$$ where $i$ is the imaginary unit. I'm having trouble with this question. I get to $a+bi$ but I have no clue how to finish it in order to determine if it converges ...
-1
votes
1answer
19 views

Find the population by the end of the same year… [on hold]

The population of a type of insect is known to be 200,000 on 1st January in a particular year. Each month, the population increases by 75,000. Find a.) the total population by the end of the same ...
-1
votes
0answers
33 views

Square root algorithm. Rudin PMA ch.3 problem 17

Fix $\alpha>1$. Take $x_1>\sqrt{\alpha}$ and define $$x_{n+1}=\dfrac{\alpha+x_n}{1+x_n}=x_n+\dfrac{\alpha-x_n^2}{1+x_n}.$$ It's easy to check that $\{x_{2n}\}_{n=1}^{\infty}$ is increasing and ...
0
votes
2answers
56 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
votes
1answer
23 views

The Sum of the first five terms of an arithmetic sequence is 65/2… [on hold]

The sum of the first five terms of an arithmetic sequence is 65/2. Also, five times the seventh terms is the same as six times the second term. Find the first term and the common difference of the ...
1
vote
1answer
40 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$