4
votes
2answers
113 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
0
votes
1answer
25 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
2
votes
1answer
37 views

Uniform convergence of $xe^{-nx}$

Does the sequence $(f_n)$ on $[0, \infty)$ given by $ f_n(x) = > xe^{-nx} $ converge uniformly? This is from Bartle's Elements of Real Analysis. I've already proven that the sequence is ...
3
votes
3answers
131 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
0
votes
1answer
30 views

Series representation of cosh or the hyperbolic cosine. [closed]

What is the Series representation of cosh or so-called the hyperbolic cosine? Can you help me guys?
8
votes
1answer
181 views

Prerequisites for understanding G.H. Hardy's 'Divergent Series'

I picked up a copy of G.H. Hardy's 'Divergent Series' a few days ago. So far I love it, as I love the ideas associated with sequences and series, but I am finding it a bit difficult to understand. I ...
4
votes
1answer
84 views

Divergent subsequence of an unbounded sequence

Let $(a_n)$ be a sequence of real numbers that is unbounded above. Show that $\exists$ a subsequence $(a_{n_k})_{k \ge 1}$ such that $\lim_{k \rightarrow \infty} a_{n_k} = + \infty$. Working ...
3
votes
0answers
43 views

Where does this series converge?

Let $ \{r_1, r_2 ,r_3,... \}$ be an enumeration of $\mathbb{Q}$. For each $r_n \in \mathbb{Q}$ define: $$u_n(x)=\begin{cases} 1/{2^n} & x>r_n \\ 0 & x \leq r_n \end{cases} $$ and let $$h ...
1
vote
1answer
131 views

Better way to prove this sequence problem relating $\lim x_{n+1}/x_n$ and $\lim\sqrt[n]{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis. Let $X= (x_n)$ be a sequence of strictly positive real numbers, let $\lim \left({ \dfrac{x_{n + 1}}{x_n}}\right) = L$, ...
1
vote
2answers
43 views

Where am I going wrong? Converging sequence

I am struck with a dilemma concerning the following exercise in Bartle's Elements of Real Analysis. Determine the convergence or the divergence of the sequence $(x_n)$ given by $$ x_n = ...
1
vote
2answers
121 views

Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence

Came across the following exercise in Bartle's Elements of Real Analysis. This is the solution I came up with. Would be grateful if someone could verify it for me and maybe suggest better/alternate ...
0
votes
0answers
43 views

On Lucas Lehmer primality Test

http://primes.utm.edu/notes/proofs/LucasLehmer.html is proof of the Lucas Lehmer Test I read. The part I do not understand is why did he consider the sequence $S_n=S_{n+1}^2-2$. I mean why would ...
1
vote
1answer
37 views

$a, z_1 \gt 0$ and $z_{n + 1} = (a + z_n)^{\frac 1 2}$ then $(z_n)$ is monotone and bounded?

Having trouble with the following exercise in Bartle's Elements of Real Analysis. Let $a, z_1 \gt 0$. Define $z_{n + 1} = (a + z_n)^{\frac 1 2}$ for $n \in \Bbb N$. Show that $(z_n)$ converges. ...
2
votes
1answer
112 views

Proof Verification: Show sequence is bounded and find limit: $x_1 \gt 1$ and $x_{n + 1} = 2 - \frac{1}{x_n}$

Came across the following exercise in Bartle's Elements of Real Analysis and am a little unsure about my solution. Would be extremely grateful if someone could verify it for me. Let $x_1 \in ...
2
votes
1answer
81 views

Help proving exercise on sequences in Bartle's Elements

Self learning Analysis and found the following exercise in Bartle's Elements of Real Analysis: Let $X = (x_n)$ be a sequence of strictly positive real numbers such that $\lim \left({\frac {x_{n ...
1
vote
1answer
33 views

to show $\sum_{i=1}^{\infty} |x_i y_i|$ converges

$X$ consists of sets of the form $(x_1, x_2, x_3, \dots)$ where $x_i \in \mathbb R$. Suppose $\sum_{i=1}^{\infty} x_i ^2$ converges. Show that : $\sum_{i=1}^{\infty} |x_i y_i$| converges. where $x,y ...
0
votes
0answers
34 views

Prove convergence of a serie little with Direct comparison test

I have the following Serie $$\sum_{k=1}^{\infty} \log(1+\frac{1}{k^2})$$ this serie should converge but when i apply the Direct comparison test it diverges $$|\sum_{k=1}^{\infty} ...
1
vote
2answers
71 views

Solution Verification: Convergence of Norms $\implies$ Convergence of Sequence

I found the following exercise in Bartle's Elements of Real Analysis. I'm learning on my own and have a couple of doubts would love it if someone could take a look. ($\left|{\left|{x}\right|}\right|$ ...
1
vote
2answers
70 views

Evaluating $\lim_{n\to\infty}\left(\frac{1-i}{4}\right)^n$

It's been a while since I have taken Calculus II so my experiences on sequences and series has gone down the drain. I'm trying to find the limit of the sequence ...
1
vote
3answers
70 views

How to find the sum of series $\sum_{i=1}^{\infty}\frac{i}{2^i}$? [duplicate]

I am learning about series of numbers at the moment. In the book there is an exercise in which I need to find the sum of : $$\sum_{i=1}^{\infty}\frac{i}{2^i}$$ I know it is equal to $2$. But how do I ...
2
votes
0answers
118 views

Unordered summation

I have a question regard unordered sumation. Definitions: Let $X$ be a set (possible uncountable) and let $f: X\rightarrow \mathbb{R}$ be a function. We say that $\sum _{x\in X}f(x)$ converges ...
0
votes
2answers
29 views

Absolute value of the difference between two sequences

Consider two sequences $(a_n)$ and $(b_n)$ both contained in $\mathbb{R}$ and assume that these two sequences satisfy $|a_n - b_n| \rightarrow 0$, then this does imply that both $a_n$ and $b_n$ are ...
0
votes
1answer
49 views

why cannot prove convergence of a serie with it's limit

i have the following serie $$ \sum_{k=1}^{\infty} \frac{{\sqrt{k} + k^3}}{{k^4+k^2}} $$ is it enough to calculate the limit to prove that it converges ? so that would be $$\lim_{k \to \infty } ...
1
vote
1answer
65 views

Rearrangement Thm

Hi everyone: In the book what I've read of analysis in the proof of the Riemann rearrangement thm there is gaps that I need to fill. There is no real problem in almost everything but for the last one ...
3
votes
1answer
120 views

Prove that $\limsup_{n \to \infty} \left(a_n + b_n \right) \leq \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$

I want to prove that for two sequences, say $a_k$ and $b_k:$ $$ \limsup_{n \to \infty} \left(a_n + b_n \right) \leq \limsup_{n \to \infty} a_n +\limsup_{n \to \infty} b_n$$ If we let $M_n =\sup \{ ...
2
votes
0answers
100 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
2
votes
0answers
187 views

Fubini's Theorem for Infinite series

In the book what I've read, there is one point where the author suggest to begin the proof of the Fubini's Theorem for infinite sum in the case when is non-negative after this try to generalize. But ...
1
vote
1answer
58 views

Testing Boundary Points Of $\sum_{n = 1}^{\infty} \frac{n!z^n}{n^n}$

I'm having some trouble testing the series indicated in the title at its boundary points. I'll sketch the preliminary work, then arrive at the problem. It is clear that the series converges absolutely ...
0
votes
1answer
42 views

Associativity of Finite Partial Sums in a Convergent Series

So I saw some topics on this, but they didn't seem to answer exactly what I was looking for. Self-learning through Understanding Analysis, a exercise is the following: So my misunderstanding is ...
3
votes
0answers
79 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
0
votes
1answer
70 views

Calculus: computation of $\sum \frac{2^i}{i!}$

$$\sum_{i=0}^\infty \frac{2^i}{i!}$$ Would anyone mind telling me what is the answer? I know this may be a silly question but I would like to know.
3
votes
1answer
52 views

How to get $\sum_{k=1}^{n-1}n\binom{n-1}{k-1}y^k \bigg(\frac{1-z^k}{k}- \frac{1-z^n}{n}\bigg)$"?

I asked a question here and got answer committing : $$(1+yz^n)(1+y)^{n-1} - (1+yz)^n=\sum_{k=1}^{n-1}n\binom{n-1}{k-1}y^k \bigg(\frac{1-z^k}{k}- \frac{1-z^n}{n}\bigg) ...
4
votes
4answers
134 views

Calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using pen and paper

How can you calculate $\sum\limits_{n=1}^\infty (n-1)/10^n$ using nothing more than a pen and pencil? Simply typing this in any symbolic calculator will give us $1/81$. I could also possibly find this ...
3
votes
2answers
95 views

Determining if a sequence of functions is a Cauchy sequence?

Show that the space $C([a,b])$ equipped with the $L^1$-norm $||\cdot||_1$ defined by $$ ||f||_1 = \int_a^b|f(x)|dx ,$$ is incomplete. I was given a counter example to disprove the statement: Let ...
2
votes
1answer
68 views

Question on the definition of infimum?

Thm: Let $K$ be compact metric space and $f:K\rightarrow \mathbb{R}$ a continuous real-valued function. Then $f$ is bounded on $K$ and attains its infimum. Since $K$ is compact and $f$ continuous ...
0
votes
1answer
31 views

Question on the convergence of $|x|^n\rightarrow 0$ when $|x|<1$.

My question is about calculating the $N$ for a convergent sequence. I want to prove that $|x|^n \rightarrow 0$ when $|x|<1$ using the $\epsilon, N$ definition. So I know that $|x^n-0| \leq ...
0
votes
1answer
78 views

Find the limit $a$ of the sequence $(a_n)_n$

I want to find the limit $a$ of the sequence $(a_n)_n$ and the smallest natural number N such that $|a_n - a| < \epsilon \forall n \geq N$ My $a_n = 5/3 - 3^n/4^n$ for all $n \in \mathbb{N},$ ...
1
vote
1answer
55 views

Prove that $a_n \times b_n \to 0$ for $n \to \infty$

I want to prove this example: If $a_n \to 0$ for $n \to \infty$ and $(b_n)_n$ is bounded. Prove that $a_n \times b_n \to 0$ for $n \to \infty$. My first guess is that I should use the definition ...
3
votes
1answer
45 views
3
votes
2answers
99 views

Please, help me to find where is a mistake in the solutions of the equation.

I have this equation and I will be very thankful to anyone who can provide me any help with the one discrepancy in my solution and the solution from the self-learning website: $$ \frac{1+\tan(x) + ...
1
vote
1answer
60 views

Summation of Modulo Sequences

I came across through simulation that multiplying and adding certain modulo sequences yield equal results. Consider the following two sequences \begin{align} g_0[k] &= \sum_{n=0}^{N-1} \left< ...
1
vote
1answer
82 views

Proof for length of period in simple modulo $N$ sequence.

I am looking for a concise proof that the length of the smallest period of the sequence $$f[n] = a n \pmod N $$ is $N$ if $(a,N) = 1$. From the Pigeonhole Principle, it is not hard to show that ...
0
votes
1answer
109 views

Relationship between different sequences generated through modulo arithmetic.

I am unsure the formal mathematical terminology/notation for dealing with sequences generated from integer modulo arithmetic. So first off, could someone recommend a book that focuses on the ...
4
votes
2answers
261 views

General question on relation between infinite series and complex numbers

This is a strictly preliminary question. I hope to elicit some discussion/s which will lead to a more acceptable form for the question on this site. I'm trying to understand how the study of the ...