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

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1
vote
3answers
78 views

Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
9
votes
2answers
162 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
6
votes
3answers
146 views

Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$

I am trying to calculate the following limit without Stirling's relation. \begin{equation} \lim_{n \to \infty} \dfrac{n!}{n^n} \end{equation} I tried every trick I know but nothing works. Thank you ...
1
vote
3answers
2k views

limits of sequences exponential and factorial: $a_n=e^{5\cos((\pi/6)^n)}$ and $a_n=\frac{n!}{n^n}$

Compute the limit $\lim\limits_{n\to\infty}a_n$ for the following sequences: (a) $a_n=e^{5\cos((\pi/6)^n)}$ (b) $a_n=\frac{n!}{n^n}$ For part (a) do I just take the limit of the exponent part and ...
4
votes
3answers
100 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 ...
2
votes
1answer
17 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 ...
1
vote
2answers
26 views

Convergence of this alternating series: $\sum_{k=0}^\infty \frac{(-1)^k}{(k+1)C^k} = C \log \frac{C+1}{C}$

I "heard" the following formula for any $C \ge 1$: $\sum\limits_{k=0}^\infty \dfrac{(-1)^k}{(k+1)C^k} = C \log \dfrac{C+1}{C}$ Is it correct? What would be a proof?
3
votes
2answers
17 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.
2
votes
3answers
60 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 ...
2
votes
4answers
89 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
57 views

Help with sequence $1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot 1$

Can anyone please provide a simplified formula for the sum of the sequence \begin{equation*} s(n) = 1\cdot n + 2(n-1) + \ldots + (n-1)2 + n\cdot1 \end{equation*} where $n$ is an integer greater than ...
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 $n→∞$ ...
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 ...
3
votes
0answers
31 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
13 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 ...
306
votes
26answers
30k views

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

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
0
votes
2answers
40 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 ...
15
votes
1answer
237 views
+100

Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?

In a gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$: Proof of equality ...
-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
47 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: $$\lim_{n \to \infty} n ((8 + \sin 2^\frac{1}{n})^\frac{1}{3} -2)$$
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
4answers
72 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
41 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 ...
3
votes
4answers
248 views

Finding the limit of a sequence $a_n=\frac{(5n^2)+n+2}{(11n^2)-3}$

What is the simplest way to find the limit of a sequence? I have this sequence $a_n=\frac{(5n^2)+n+2}{(11n^2)-3}$ and I have to find its limit. However I'm lost on how to do this.
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 ...
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: ...
0
votes
1answer
541 views

Convergence of Monotone sequences? example

An example of an unbounded increasing sequence that satisfies the assumptions of the convergence of monotone sequences...? According to the convergence of monotone sequences if a sequences is ...
-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
22 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 ...
4
votes
4answers
229 views

Solve $\sum_{i=1}^{\infty}S^i=1$ for $S$

$\sum_{i=1}^{\infty}S^i=1$ for S. So I'm assuming that they assign S to be 1 and then you just keep increasing the exponent as the term number goes up till infinity. But that would just be 1+1+1... ...
8
votes
1answer
236 views

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such 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 $\liminf a_n \le \liminf B_n \le \limsup B_n \le \limsup a_n$. ...
21
votes
5answers
2k views

Is there any explanation for the repetitions after decimal point on divisions like 24/7

I was trying to divide 24 by 7 using a pen and a paper. After I had no more space on my checkerboard paper, I decided to put it on a calculator. The calculator returned 3.428571428571429 and I ...
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 ...
10
votes
1answer
176 views

Sum of Harmonic numbers $\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$

Finding the closed form of: $$\sum\limits_{n=1}^{\infty} \frac{H_n^{(2)}}{2^nn^2}$$ where, $\displaystyle H_n^{(2)} = \sum\limits_{k=1}^{n}\frac{1}{k^2}$ It appears when we try to determine the ...
0
votes
1answer
85 views

Sequence $ x_n=(1-\frac{1}{n})\sin (n\pi/3)$lim sup, lim inf

Let $ x_n=(1-\frac{1}{n})\sin (\frac{n\pi}{3})$ for n $\ge $1. denote l=limit inferior and s=limite superior Then -$\sqrt{3}/2\le l\lt s \le \sqrt{3}/2$ -$1/2\le l\lt s \le 1/2$ $l=-1 $,$ s=1$ ...
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 $\mu(A) = \sum_{a \in A} \frac{1}{a}$. ...
6
votes
1answer
295 views

Integral of the ratio of two exponential sums

I am trying to find a lower bound on the following integral \begin{align*} \int_{y=-\infty}^{y=\infty} \frac{ (\sum_{n=[-N..N]/\{0\}}n e^{-\frac{(y-cn)^2}{2}})^2} {\sum_{n=[-N..N]/\{0\}} ...
4
votes
3answers
373 views

derive the Maclaurin series by using partial fractions.

derive the Maclaurin series for the function $(x^3+x^2+2x-2)/(x^2-1)$ by using partial fractions and a known Maclaurin series. question. how can I use partial fractions in this case? Is this case the ...
0
votes
1answer
17 views

Sequence of holomorphic functions and approximation by polynomials.

Let $\Omega=\{ z\in \mathbb{C}:$ $Im$ $z>0,$ $|z|>1\}\cup\{z \in \mathbb{C}:$ $Im$ $z<0$ $|z|>1\}$ I know that since $\hat{\mathbb{C}}\setminus \Omega$ is connected there's a sequence of ...
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
0answers
72 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 ...
4
votes
4answers
196 views

Find the Exact sum

Give the fourier series representation of $f(x) = x$ on $[-\pi, \pi]$. Use the result to give the exact sum of... $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}$$ $$\text{ where } x \in [-\pi,\pi]$$
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 ...
-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
53 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!
0
votes
1answer
35 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 $a_1$?