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

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0
votes
1answer
21 views

Find the value of $\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$.

Find the value of $$\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$$ I used WolframAlpha obtaining $$\psi^{(0)}(2n)-\frac{1}{n}+\gamma$$ where $\gamma$ is the Euler-Mascheroni constant and ...
0
votes
3answers
42 views

How to prove that the sum of a convergent geometric series of the form $1 + r + r^2 … + r^n > 1/2$?

I am trying to prove that the sum of a convergent geometric series of the form \begin{equation*} 1 + r + r^2 .... + r^n > \frac{1}{2} \end{equation*} but I have no idea how to go about this. ...
0
votes
2answers
21 views

Establishing a trigonometric identity for $n\in\mathbb{N}$

The original problem was showing that this infinite sum converges to $\tan\theta$: $$\sum_{n=1}^\infty \frac{\tan\dfrac{\theta}{2^n}}{\cos\dfrac{\theta}{2^{n-1}}}$$ One hint was given: the series ...
2
votes
1answer
19 views

Radius of convergence of a power series with $a_n$ convergent

Let $\{a_n:n \geq 1\}$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_n$ is convergent and $\sum_{n=1}^{\infty} |a_n|$ is divergent. Let $R$ be the radius of convergence of the power ...
3
votes
2answers
20 views

Determine whether series converges or diverges

$$\sum_{n=1}^{\infty}\frac{\sin\left(\frac{5\pi}{3}n\right)}{n^{\frac{5\pi}{3}}}$$ Hello, I thought about using Squeeze Theorem but the 5π/3 threw me off. Thanks in advance.
0
votes
2answers
31 views

Can one prove divergence by showing a series has at most one solution for an=0?

Say I have any series, I would think it was enough to show that this series equals 0 at most once to prove it diverges. My logic is, For a series: $\sum a_n →∞$, and diverges, if $a_n≠0$ for $n→∞$ ...
2
votes
3answers
61 views

Math about Geometric series

In a geometric series, the sum of $1^{st}$ term $+$ $2^{nd}$ term $+$ $3^{rd}$ term $= 38$, the sum of $2^{nd}$ term $+ 4^{th}$ term $= 17 \frac{1}{3}$; how to calculate the common ratio? ( it is ...
1
vote
0answers
37 views

Uniformly convergence question

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
4
votes
0answers
33 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
1
vote
0answers
14 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
2
votes
4answers
91 views

Compute $\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$

How do I evaluate the following limit? I guess I should do a comparison, but I've got no clue about what to do. Could you give me a hand? $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + ...
0
votes
2answers
41 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
4
votes
3answers
101 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
-4
votes
2answers
61 views

Why is $n^\sqrt n - 2^n \to - \infty$ and $\sqrt n ^n - 2^n \to +\infty$ [on hold]

Can you explain technically why the following limits are correct? $$n^\sqrt n - 2^n \to - \infty$$ and $$\sqrt n ^n - 2^n \to +\infty$$
-1
votes
2answers
48 views

How do I calculate the following limit? $\lim_{n \to \infty} n ((8 + \sin (2^\frac{1}{n}))^\frac{1}{3} -2)$

How do I calculate the following limit? I'm short on ideas for this one: $$\lim_{n \to \infty} n ((8 + \sin 2^\frac{1}{n})^\frac{1}{3} -2)$$
3
votes
4answers
74 views

How do I calculate $\lim_{n \to \infty} n^\frac{1}{n} (n+1)^\frac{1}{n+1} … (2n)^\frac{1}{2n}$

How do I calculate the limit of the following sequence? $$\lim_{n \to \infty} n^{\frac{1}{n}} (n+1)^\frac{1}{n+1} ...... (2n)^\frac{1}{2n}$$
0
votes
1answer
12 views

Find limit inferior and limit superior of $[1+\sin n]$ and $n - [\sqrt n]$

I have to find the limit inferior and limit superior of the following sequences: $$[1+\sin n]$$ and $$n - [\sqrt n].$$ I have done similar exercises before, but never with the integer part function ...
2
votes
3answers
42 views

Prove $\lim_{n\to\infty} \left(\sin \frac{\pi n}{2} \cdot \cos \left(\sin \frac{1}{n} \right ) \right)$ doesn't exist

I must prove that the limit $$\lim_{n\to\infty} \left(\sin \frac{\pi n}{2} \cdot \cos \left(\sin \frac{1}{n} \right ) \right)$$ doesn't exist and also find all of its partial limits. Apparantely ...
2
votes
1answer
17 views

Sequences formed by integer evaluations of polynomials modulo $ p^{k} $, where $ p $ is a prime number and $ k \in \Bbb{N} $.

I have the following question. Let $ p $ be a prime number and $ k $ a positive integer. Let $ (a_{n})_{n \in \Bbb{Z}} $ be a two-way sequence in $ \Bbb{Z} / p^{k} \Bbb{Z} $. Then is it true that ...
3
votes
2answers
32 views

Limit of cos function in a sequence

In my assignment I have to calculate to following limit. I wanted to know if my solution is correct. Your help is appreciated: $$\lim_{n \to \infty}n\cos\frac{\pi n} {n+1} $$ Here's my solution: ...
-6
votes
3answers
28 views

Working out a sequence from later terms [on hold]

Question on a sequence: 50th term is 349, 51st is 354, 52nd is 359. Find 1st term and 100th? I really need help to get started on this. Please can you explain in a less complex way im only a ...
1
vote
0answers
24 views

Prove that one eight of Gauss circle problem counts integer sided acute triangles with largest side n.

In the OEIS there is the sequence A247588 starting: 1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27,... It has the name "Number of integer sided acute triangles with largest side n." Let $a(n)$ be that ...
1
vote
1answer
29 views

Prove a limsup and liminf inequality.

Let $(a_n)_{n\in\mathbb{N}}$ be a bounded sequence and let $B_n = \frac{1}{n} \sum_{i=1}^n a_i$ for each $n \in \mathbb{N}$. Prove that $\liminf a_n \le \liminf B_n \le \limsup B_n \le \limsup a_n$. ...
0
votes
2answers
65 views

Solve $x^2=\cos x$ using Taylor series for cosx

I have the following equation:$x^2=\cos x$ and calculating the Taylor series of $3rd$ degree around $0$ I've got: $x\approx \pm\sqrt{\frac{2}{3}}$ However, now I need to prove that if x is a ...
0
votes
0answers
14 views

Math Software for a Identifying/Generating Sequences [on hold]

I'm looking for (preferably Windows or DOS) software that I can feed number sequences that I've designated as "good", and possibly number sequences that I've designated as "bad" -- although the latter ...
0
votes
3answers
48 views

Is there another way to solve this Trigo in series? [duplicate]

Find the value of $$\cos ^2\theta+\cos^2 (\theta+1^{\circ})+\cos^2(\theta+2^{\circ})+...... +\cos^2(\theta+179^{\circ})$$ Attempt, $$\cos x=-\cos(180^\circ-x),\sin x=\cos(90^\circ-x),\cos ...
6
votes
2answers
393 views

Integrating over the naturals. What does it mean?

Let $F$ be the power set of $\Bbb{N}$ and consider the measurable space $(\Bbb{N}, F)$. Then what does it mean to take the integral with respect to the measure $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
1
vote
2answers
52 views

Show that this series is divergent using the comparison test.

Let $\langle a_n\rangle$ be a sequence of positive numbers. Consider $\sum_{n=1}^\infty{Y_n}=\sum_{n=1}^\infty{\left(\frac{a_n}{n}+\frac{n}{a_n^2}+\frac{a_n}{n^3}\right)}$. Show this diverges using ...
4
votes
1answer
83 views

The infinite series $\sum_i a_i \prod_{j=0}^{i - 1}(1 - a_j)$ has sum equal to $1$

Suppose we have an infinite sequence of constants $\{ a_i \}_{i=0}^{\infty}$, where $0 < a_i < 1$ for every $i$, and we define $b_i \equiv a_i \prod_{j=0}^{i - 1}(1 - a_j)$. How do we prove ...
-6
votes
0answers
35 views

Help Please, If anyone solve this I'll thankfull to him [duplicate]

find the limit of the following series 1+1+3/4+1/4+5/16+3/16+7/64+5/64+.... Thanks
2
votes
3answers
54 views

A series involving harmonic numbers

Does anyone know the exact value of this: $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k}{k} $$ or this: $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(2)}}{k} $$ Thanks!
2
votes
1answer
37 views

On a property of Fourier coefficients

I need to prove the following: If $(\Phi_n)_{n\ge0}$ is an orthonormal system of integrable functions defined on some interval $[a,b]$, and $(c_n)_n$ is a sequence of reals such that $\sum c_n ...
9
votes
0answers
105 views

Product of $n^n$

Is there a formula that defines $$(1^1)(2^2)(3^3) . . . (n^n)?$$ Most of the texts on the internet tackle series with the same exponent, but how about this one? Sorry for my mistakes
-10
votes
1answer
65 views

I want you to help me with this question please [on hold]

find the limit of the following series $1+1+3/4+1/4+5/16+3/16+7/64+5/64 +....$ Simple prove for the question $1111-22=11(101)-11(2)=11(101-2)$ $=11(99)=(11^2)(3^2)=33^2$
0
votes
1answer
36 views

“Standard first term” of a series

A (not so interesting) question. Just to get good practices. When a series is considered in English, is the first term usually $a_0$ or $a_1$?
3
votes
0answers
35 views

Number of squarefree numbers and the Basel problem

Who discovered/proved that there are about $$ \frac{x}{\zeta(2)} $$ squarefree numbers up to $x$, or (roughly) when was this first known? Today I think this is considered 'obvious', but I don't know ...
0
votes
2answers
35 views

Prove or disprove if a series is convergent that it implies the square of the series is convergent as well

Heyy, Can we prove or disprove the following $$ \Sigma a_n \text{ is convergent} \Rightarrow \Sigma a_n^2 \text{ is convergent} $$ Since the statement cannot be proven without knowing whether the ...
2
votes
1answer
43 views

Show that the binomial series satisfies: $(1+x)f'(x)=\alpha f(x)$

If $f(x) = \sum_{n=0}^{\infty}{\alpha\choose n}x^n$, show (without assuming the results of the binomial theorem) that $$(1+x)f'(x)=\alpha f(x)$$ for $|x|<1$ I've already shown that the sum ...
1
vote
1answer
28 views

Finding value (Trigo Series) [duplicate]

Find the value of $$\cos ^2\theta+\cos^2 (\theta+1^{\circ})+\cos^2(\theta+2^{\circ})+......+\cos^2(\theta+179^{\circ})$$ Can anyone teach me where to start with? I've no idea.
1
vote
0answers
21 views

Find $Z$ transform of given signal

Given the discrete signal $h(n)=r^n\frac{\sin{[(n+1)\theta]}}{\sin{\theta}}$ if $n \geq 0$ and $h(n)=0$ otherwise, find the $Z$ transform of $h(n)$. What I did: We know that ...
10
votes
1answer
120 views

Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$

As a follow up of this nice question I am interested in $$ S_1=\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2} $$ Furthermore, I would be also very grateful for a solution ...
0
votes
1answer
23 views

How to get this very simplified demographic forecast?

I'm working on the simulation of a population growth. The variables and hypothesizes are the following: Lifetime: X years (X constant for everybody, yeah !) Initial population: Y people (with always ...
0
votes
3answers
44 views

If $X$ has a Poisson distribution with $E[X]=\lambda$, does $Var[X^2]=4\lambda^3+6\lambda^2+\lambda$?

Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I ...
-9
votes
0answers
42 views

Limit of a particular series [on hold]

Please help me find the limit of the following series: $$1+1+3/4+1/4+5/16+3/16+7/64+5/64+\cdots$$ this is simple prove for it 1111-22=11(101)-11(2)=11(101-2) =11(99) =(11^2)(3^2) =33^2
0
votes
2answers
24 views

Recursive formula of an explicitly defined sequence

Does there exist a recursive formula for this sequence? $$a_n=\frac{2}{3}\left(1-\left(-\frac{1}{2}\right)^n\right), n\in\mathbb{N}_0$$
2
votes
1answer
46 views

Prove that all the five sequences converge to the same point $P \in \mathbb{R}^3$.

Let five sequences $A_n, B_n, C_n, D_n, E_n \in \mathbb{R}^3$ be constructed as follows: $A_0, B_0, C_0, D_0$ and $E_0$ are some given points of the space and $A_{n+1}, B_{n+1}, C_{n+1}, D_{n+1}, ...
2
votes
1answer
34 views

How many Fibonacci Numbers are in the sequence

I have $I_n = \{2^n + 1, 2^n + 2, 2^n + 3, \dots , 2^{n+1}\}$ and I am trying to prove using induction how many Fibonacci numbers are there. First, the length of $I_n$ is $|I_n| = 2^n$ then for $F_0 ...
3
votes
3answers
48 views

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$

$\lim \limits_{n \to \infty}$ $\prod_{r=1}^{n} \cos(\frac{x}{2^r})$ How do I simplify this limit? I tried multiplying dividing $\sin(\frac{x}{2^r})$ to use half angle formula but it doesnt give ...
2
votes
1answer
56 views

Looking for formula of $\sum_{k=1}^m (-1)^k \dfrac {x^2(x^2-1)…(x^2-k+1)}{(x+1)(x+2)…(x+k)}$

Let \begin{equation*} u_k:=(-1)^k \dfrac {x^2(x^2-1)...(x^2-k+1)}{(x+1)(x+2)...(x+k)}. \end{equation*} Can we find the sum of first $m$ of $u_k$ 's? That is, is there any formula for $\sum _{k=1}^m ...
0
votes
1answer
25 views

How to calculate the number of combinations of $x$ integers, each with a value between $y$ and $z$?

For example, if I have 4 integers, and each can be between 0 and 36, how many combinations are there? If the numbers have appeared before, but in a new order, then this still counts as a new ...