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

learn more… | top users | synonyms (5)

-1
votes
1answer
35 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
28 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 $f(x_{n})$ $\rightarrow f(x)$. ...
3
votes
0answers
41 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 ...
3
votes
4answers
803 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
154 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
94 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
votes
0answers
14 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
32 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 ...
0
votes
2answers
69 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
votes
1answer
28 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}$ Any ideas on how to proceed? Or are there ...
-1
votes
1answer
80 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
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 ...
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 ...
0
votes
1answer
66 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
85 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
vote
0answers
47 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
22 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
58 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 $j\in$ ...
0
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
80 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
42 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
32 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
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
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
35 views

Approximaing Gamma function

For $c>1$ and $0<\theta<1$, we wish to approximate (upper bound) following Gamma function: $$\int_c^{c\theta}x^{-3}e^{-x}dx $$
1
vote
2answers
65 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
1answer
33 views

Geometric Progression sums and sums of squares

Sum of the first $4$ terms in GP is $30$ and the sum of their squares is $340$. Find the numbers. How do I solve this?
0
votes
0answers
25 views

Chance of wining Uno 7 Times in a row

I am Intrigued to determine the odds of winning such a game with 4 players.. and in 7 times consecutively Would be great to have a clear answer on this Thks Phil
2
votes
1answer
82 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 ...
1
vote
0answers
21 views

Find first positive perfect square in polynomial time

I have a quadratic. for example $$1x^2+6884x+3297$$ Is it possible to find the first perfect square in the series in polynomial time where both x and y are whole positive integers. In the above ...
4
votes
2answers
51 views

Limiting value of $\frac{x^n e^x}{n!}$ as $n\to\infty$

For the Taylor Series the remainder is of the form $$R_n = \frac{(x-a)^n}{n!} f^{(n)}(\xi) $$ with $a \leq \xi \leq x$ For the series of $e^x$ about $0$ (that is, the Maclaurin series) the remainder ...
0
votes
0answers
15 views

Does this property of time dependent sequences have a name?

For $i\in \mathbb{N}$ let $ \chi_i \colon \mathbb{R}^+ \to \mathbb{R}$ be such that for each $t\in \mathbb{R}^+$ we have $\chi_i(t) \in \mathcal{l}_2$ Suppose further that ...
12
votes
3answers
565 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. ...
5
votes
1answer
54 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
1
vote
1answer
39 views

limit of sum $\dfrac{(-1)^{n-1}}{2^{2n-1}}$

What is: $$\sum^{\infty}_{n=1}\dfrac{(-1)^{n-1}}{2^{2n-1}}$$ I have done a Leibniz convergence test and proved that this series converges, but I do not know how to find the limit. Any suggestions?
2
votes
2answers
36 views

What function is this? $\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$ [on hold]

I am interested to know what function represents the following series: $$\sum_{k=0}^\infty \frac{2^{2k}z^{2k-1}}{(2k)!}$$
0
votes
2answers
29 views

Method of finite differences solutions help?

For homework we were given this sequence: -2 8 27 85 260 ____ 2365 And asked to find the number in the blank. Well, I got the ...
0
votes
2answers
33 views

Sequences and Series - Find the value of n for which…

I am having some difficulty trying to solve this question. I have been given this question - Find the correct value of the letter n for which Xn = 5n - 2 and ...
3
votes
0answers
39 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
3
votes
3answers
103 views

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$? It is well-known that $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}$, so ...
5
votes
2answers
124 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{\zeta(1-n)(-1)^{n+1}}{2^{n-1}}$$ $B_n$ are the Bernoulli numbers. Mathematica does not help.
0
votes
1answer
51 views

What is the sum of this series? $\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$ [duplicate]

I want to know what is the sum of this series: $$\sum_{n=1}^\infty\frac{2^n (-1)^{n+1} B_n}{n}$$ Mathematica does not help.
1
vote
2answers
141 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 ...
2
votes
1answer
57 views

Algorithm for computing square roots.

Fix a positive number $\alpha$. Choose $x_1>\sqrt{\alpha}$ and define $x_2, x_3, x_4, \dots$ by the recursion formula $$x_{n+1}=\frac{1}{2}\left(x_n+\frac{\alpha}{x_n}\right).$$It's easy to check ...