For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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3
votes
3answers
152 views

Summing infinitely many numbers: how to assign a value?

If we take $S = 1-1+1-1+1-1+1-1+...$ we can show (in many different ways) that the result of the sum is $\frac{1}{2}$. One way for example would be to add $S$ to itself but shift it along one place, ...
1
vote
2answers
50 views

Closed form sum for the series given below?

Does the following series have a closed form sum? $$f(n,r) = \sum_{i=0}^n \binom{r+i}{r}$$
1
vote
3answers
47 views

Why is $\sum\limits_{i=0}^{n}(r-1)^{n-i}{n\choose i} = r^n$?

I was solving a problem and found that $\sum\limits_{i=0}^{n}2^{n-i}{n\choose i} = 3^n$. So I tried to generalise it and got $\sum\limits_{i=0}^{n}(r-1)^{n-i}{n\choose i} = r^n$. Is it true for $r ...
5
votes
1answer
147 views

Computing $\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\cdots $

What tools would you recommend me for computing this series? $$\int_0^{\large1/1^2} \left\{ \frac{1}{t} \right\}\,\mathrm{d}t+\int_0^{\large 1/2^2} \left\{ \frac{1}{t} ...
2
votes
1answer
27 views

$\sum_{n=1}^{\infty} {(1-\cos(\sin 1/n))}^{w}$ with $w$ as parameter

Let $f(x)=(1-\cos(\sin x))$; $a_n=f(1/n)$ for $n\in\mathbb{N}$ For which $w>0$ series $$\sum_{n=1}^{\infty} {a_n}^{w}$$ converge? I haven't got a slicest idea how to check that, absolutely none ...
0
votes
1answer
67 views

Closed form of the series

I want to evaluate $\sum_{i=1}^{n} (x+i)^4$ So what i did is, after expanding it i reduce it to following form $ x^{4} * n + 4 x^{3} * \sum_{i=1}^{n}i + 6x^2\sum_{i=1}^{n}i^{2} + ...
1
vote
0answers
60 views

What is the most “powerfull” method to prove a sequence is increasing or decreasing?

Given a sequence $a_n$ defined in a recursive manner, the methods I know to prove if the sequence is increasing are: 1) observe if $a_{n+1} - a_n > 0 \ \forall n.$ 2) take $\frac{a_{n+1}}{a_n}$ ...
2
votes
1answer
115 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
3
votes
3answers
212 views

Monotonicity of the sequences $\left(1+\frac1n\right)^n$, $\left(1-\frac1n\right)^n$ and $\left(1+\frac1n\right)^{n+1}$

I am working on the following sequences. $$x_n=\left(1+\frac1n\right)^n \qquad z_n=\left(1-\frac1n\right)^n \qquad y_n=\left(1+\frac1n\right)^{n+1}$$ I am trying to prove that $x_n$ and $z_n$ are ...
3
votes
2answers
70 views

Sign of a series

Someone could compute the sign of the following series ? \begin{equation} \underset{k > 0}{\sum} \frac{\sin (kx)}{k} \end{equation} I expect that is the same as the first term $\sin x$ because of ...
1
vote
1answer
63 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
0
votes
2answers
42 views

Sequence converging to zero

Prove the sequence $x_n = \frac{2n}{n^2 + 1}$ converges to zero. Attempt proof: $x_n = \frac{2n}{n^2\left(1 + \frac1{n^2}\right)} = \frac2{n\left( 1 + \frac1{n^2}\right)}$ Now we can know $\frac2n ...
1
vote
2answers
65 views

on the exercise 8.10 Apostol. (limit of sequence)

The exercise states: prove that the limit of the sequence $$a_{n+2}=(a_na_{n+1})^{1/2} \ where \ a_1 \ge 0, a_2 \ge 0 $$ is $L = (a_1a_2^2)^{1/3}$ The solutin says: $$Let \ b_n = ...
0
votes
1answer
30 views

Assumption in a convergence proof

Im in the middle of a proof of the fact that for a>0, if lim $x_n$ = a, then lim $\sqrt{x_n}$ = $\sqrt{a}$. I'm in the step that i use $| \sqrt{x_n} - \sqrt{a} |$ = $ \frac{|x_n - a|}{ ...
28
votes
3answers
2k views

Is $\sum_{n=1}^\infty \frac{\sin(2n)}{1+\cos^4(n)}$ convergent?

As I was just checking this 'child prodigy' out on Youtube, I stumbled upon this video, in which Glenn Beck asks the kid to do the following proof: Further on, the kid starts sketching a proof ...
2
votes
0answers
47 views

Evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ where $n,k$ are fixed

Is there a general way/technique to evaluate $ \sum\limits_{j \in \mathbb{Z}_{\geq 0}} {n \choose r+kj}$ in terms of $r$, where we consider $n$ and $k$ fixed natural numbers and $n > k$? (here, ...
1
vote
1answer
56 views

Rules for constructing a sequence (induction)

What steps are required to built a sequence? Here is an example: Show that there is a sequence of rational numbers that converge to pi. Note that $0 < \pi$. Since the rationals are dense in the ...
0
votes
1answer
70 views

Lim sup ratio test and Ramifications

Let $x_n$ be a real number sequence. I have managed to prove that,for a sequence $x_n$ of positive terms : if lim sup $(x_{n+1}/x_{n}) < 1$ then $x_n$ is not only eventually monotonicly ...
7
votes
0answers
139 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
1
vote
3answers
71 views

Is $(n^{1/n}-1)\in O(n^{-\frac12})$ as $n\to \infty$?

Let $$x_n=n^{1/n}-1$$ An exercise asks me to prove that $\{x_n\}\to 0$ and that $x_n\in O(n^{-\frac12})$ as $n\to \infty$. Now, I could easily prove the first part. But to me $x_n\notin ...
9
votes
1answer
108 views

Consecutive terms in Pascal's Triangle

is it known whether or not there are infinitely many pairs of consecutive terms in this sequence: http://oeis.org/A006987 ? The sequence is the list of numbers expressible in the form ...
0
votes
0answers
40 views

A basic sequence and series question

Consider the series $$\sum_{n=1}^{\infty}z_n=\sum_{n=1}^{\infty}(x_n- y_n)$$ both $x_n$ and $y_n$ are non-negative? Assume that for the above series the following two are only possibilities (i.e if ...
0
votes
1answer
50 views

Arithmetic sequence: determine number of addends

Given that the sum of first 5 terms of an arithmetic sequence is 65, the sum of the last five terms is 1090,and the sum of all the terms is 5313.Find the number of terms in the sequence I try to ...
2
votes
2answers
86 views

Proving the convergence of $a_{n+2}=(a_{n+1}a_n)^{1/2} \qquad (a_1\ge0, a_2\ge0)$

I am trying to solve exercise 8.10 from Apostol-Mathematical Analysis. I need to prove that the following sequence converges to $L=(a_1a_2^2)^{1/3}$. $$a_{n+2}=(a_{n+1}a_n)^{1/2} \qquad (a_1\ge0, ...
36
votes
6answers
1k views
1
vote
0answers
223 views

Double summation of a geometric series

I am interested in the following sum for a given value of n: $ \sum\limits_{x=1}^{n} \sum\limits_{y=1}^{n}x^y$ I can simplify this to $ \sum\limits_{x=1}^{n} \frac{x^{n+1} - x}{x - 1}$ From here ...
4
votes
1answer
116 views

Exam exercise on sequence $a_n = \sin(n)$ [duplicate]

Prove that the sequence $a_n = \sin(n)$ cannot converge when $n \rightarrow \infty $ I tried to find two subsequences that converge to different values but I am having trouble with the fact that $n ...
0
votes
3answers
48 views

If $x_n \to \infty$ and $y_n$ Cauchy, then $x_n + y_n \to \infty$

Suppose that $(x_n)$ is a sequence of real numbers such that $\displaystyle{\lim_{n \to \infty} x_n = \infty}.$ Suppose also that $(y_n)$ is a Cauchy sequence of real numbers. Show that $$\lim_{n \to ...
0
votes
0answers
32 views

A sequence of intervals- Trying to find a fixed point -

This might be a trivial question but I couldn't come up with a clever trick,theorems or whatnot. Suppose $I_0=\left[\frac{1}{h_0},\frac{1}{l_0}\right]$ where $h_0=1$ and $l_0=\frac{1}{2}$. Given ...
1
vote
1answer
47 views

Generalized series expansion for $\Gamma(a,z)$ in $a$ at $a=0$

I need the generalized series expansion of $\Gamma(a,z)$ for $z\in\mathbb{C}$ and $a\to0^+$. Mathematica gives a result that seems to be correct, but I have to make sure of its validity. I came ...
1
vote
1answer
41 views

Exist a function that satisfies these conditions?

Exist a function that safisties...? $$f(n)=\sum_{k=1}^{n}g(k)=n\ : g(k)\in (0,+\infty),\ g'(k)> 0, n\in \mathbb{N}$$ And this...? $$f(x)=\int_{0}^{x}g(t)dt=x\ : g(t)\in (0,+\infty),\ g'(t)> ...
2
votes
1answer
141 views

Uniform integrability of a sequence of functions.

I am considering the following sequence of functions. I think it converges pointwise to $0$ because the intervals in the domain in which the nth function is greater than zero eventually shrink and ...
2
votes
3answers
54 views

How can I prove this? If $\lim a_{2n}=\lim a_{2n+1}=L\in \mathbb{R}$ then $\lim a_{n}=L$

I am trying to solve this problem which seems obvious, but I am stuck I need a hint to start. Prove that if $\lim a_{2n}=\lim a_{2n+1}=L\in \mathbb{R}$ then $\lim a_{n}=L$, for any sequence ...
1
vote
1answer
75 views

Summation of: $\sum_{1}^{\infty}\left(\frac{2}{3}\right)^x$ [duplicate]

This is a subsection in my statistics homework. It goes back to calculus II and summations, and it's been a long time since I've studied it so I'm rusty. I'm looking to solve the summation of ...
0
votes
2answers
73 views

Proof of the inequality: $\sum_{k=1}^N\frac{1}{k^N}\gt\frac{N}{\Gamma(N+1)}$.

The following inequality seems to hold: $$\forall N\gt2,\displaystyle\sum_{k=1}^N\dfrac{1}{k^N}\gt\dfrac{N}{\Gamma(N+1)}$$ Is it possible to prove it analytically? Thanks.
0
votes
3answers
88 views

Sequences and Series - AP and GP

Question: If a,b,c are in GP and $$a^{1/x} = b^{1/y} = c^{1/z}$$ prove that x,y,z are in AP I tried writing b and c in terms of a, by assuming a common ratio r, however, I was unable to proceed from ...
3
votes
1answer
178 views

How to compute this sum?

I want to sum the following: $$f(n) = \sum_{i=1}^n (i^3 \cdot (n \mod i))$$ Since the sum can be huge I have to output the sum modulo some given number m. How can I approach this problem? Also, n ...
3
votes
2answers
89 views

First-year analysis question: an increasing sequence

My student showed me this question and I'm genuinely having trouble with it. I will list all the parts so that the pre-requisites are clear. Q. Let $c \in (0, 1/2)$. We define a sequence $(a_n)$ by: ...
0
votes
1answer
52 views

How many terms are required to get $D$ digits of Riemann zeta prime function?

How many terms are required to get $D$ digits of Riemann zeta prime function $\zeta_p(s) = \sum_p \frac{1}{p^s}$? Sebah & Gourdon mentions that finding $\zeta_p(2)$ to 20 digits by using $\sum_p ...
0
votes
1answer
295 views

provide a combinatorial proof that $C_{n+1} = C_0C_n + C_1C_{n-1} + …. + C_kC_{n-k} + …C_nC_0$

(a) Let $C_n$ denote the number of ways of writing a valid list of open and closed parentheses of length $2n$ (valid means that at any point along the list, the number of open parentheses must be ...
4
votes
2answers
58 views

How to prove the series converge for every $\theta$?

How to prove the following series converge for every $\theta$, any suggestions are welcome? $$ \sum_{n=1}^{\infty}\frac{(-1)^n\sin(n\theta)}{n}.$$
4
votes
2answers
147 views

Is the series $\sum_{1}^{\infty}\frac{1}{p_{j}}$ where $p_{j}$ is the $j$th prime convergent? [duplicate]

Does the series $\sum_{n=1 }^{\infty}1/p_{j} $ of reciprocal primes converge? Experimentally, it seems convergent.
0
votes
1answer
38 views

Clarifying the meaning of uniform convergence

I just want to clarify the notion of uniform convergence. Suppose, I have a sequence of function like this, $$f_n(x) = \frac{n}{e+n^2x^2}$$ Clearly for $|x| > 0$, $f_n(x) \to 0$ if $n \to ...
4
votes
5answers
180 views

Sums $\sum_{k = 0}^n k^t {n \choose k}$ where $t$ is a positive integer

I recently came across the problem of finding out the sum $\sum_{k = 0}^n k^2 {n \choose k}$. The solution that I've found goes something like this: $\sum_{k = 0}^n k^2 {n \choose k}=\sum_{k = 0}^n ...
4
votes
1answer
93 views

On a recursive sequence (exercise 8.9 Apostol)

The exercise states: show convergence of the sequence ${a_n}$ knowing that: $$|a_n| \le 2, \ \ \ |a_{n+2}-a_{n+1}| \le \frac{1}{8}|a_{n+1}^2 - a_{n}^2|.$$ The solution states: $$|a_{n+2}-a_{n+1}| ...
2
votes
6answers
148 views

Does the sequence converge, and to what? [closed]

We have a sequence $\{a_n\}$ $$a_0 = 0$$ $$a_{n+1} = \frac{a_{n}}{2} + 1$$ Does it converge? And to what?
0
votes
1answer
61 views

Proof of a property on a Fibonacci like sequence

The recurrence relation of the series is the following, $N(1) = N(2) = N(3) = 1$ $N(n) = N(n-1) + N(n-3)$ for $n>3$ I need to prove by induction on $a$ that, $N(n) = N(a+2)N(n-1-a) + ...
3
votes
1answer
109 views

exam exercise on Series problem.

The exercise states: Does the series $$\sum\limits_{n=1}^\infty \int_{0}^1 \frac{x^n dx}{x+1}$$ converge? The solution states as the first step: $$I_n =\int_{0}^1 \frac{x^n dx}{x+1} $$ then $ ...
1
vote
2answers
73 views

Is this series divergent or convergent?

I've been stuck with this problem for a couple of days trying to solve it but got no where till now. The problem states that we have to prove if the series given below is convergent or divergent, if ...
2
votes
1answer
50 views

Testing convergence of the series of $n^p((n-1)^{-1/2}-n^{-1/2})$

Exercise 8.15 (l) of Analysis by Apostol states: Test for convergence: $$\sum\limits_{n = 2}^\infty n^p\left(\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}} \right)$$ The solution I have states as the first ...