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

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0
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4answers
31 views

Proof by induction: $\sum_{k=3}^n k = (n-2)(n+3)/2$

I'm trying to prove the following by induction: $$\sum\limits_{k=3}^n k = \frac{(n-2)(n+3)}{2}$$ Any Ideas about finding the base and induction case?
2
votes
1answer
20 views

Show that $\lbrace 1-\frac{1}{n} \rbrace_n$ does not converge in the Sorgenfry topology.

Consider $\{1-\frac{1}{n}\}_n=\{0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots\}$. If $\{1-\frac{1}{n}\}_n$ converges, then $\{1-\frac{1}{n}\}_n \rightarrow 1$. If it converges, then, by definition, ...
1
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2answers
20 views

What is the correct notation for a sub-sequence?

I want to say that a sequence is a subsequence of $A_n$ for all even indexes, is it valid to write it as $$ k\in \mathbb{N},\, n > k$$ $$(B_k){\,^\infty_{k=1}} = (A_n){\,^\infty_{n=2k}} $$ or $$ ...
1
vote
2answers
38 views

$\sum^{\infty}_{n=1}(n-\sqrt n)/(n^{2}+5n)$ diverges

Show that $$\sum^{\infty}_{n=1}\frac{n-\sqrt n}{n^{2}+5n}$$ diverges. I have tried Root test, ratio Test, Cauchy condensation Test but all have failed. I think this has to be done by Comparison Test ...
0
votes
1answer
18 views

Cesàro summable sequences

During some homeworks the following question came into my mind (it is not part of the homeworks): Let $(a_k)_{k \in \mathbb{N}}$ be a Cesàro summable sequence in $\mathbb{C}$ and let $a := \lim_{n ...
0
votes
1answer
22 views

sum of series independent of variable in sequence

\begin{equation} -\frac{k^2+4}{k^2-8}-\frac{12k^2}{(k^2-8)^2}\sum\limits_{n=0}^{\infty} (\frac{k^2}{k^2-8})^n=\frac{1}{2} , -2\leq k\leq 2 \end{equation} This is an equation which sums up to a value ...
1
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3answers
47 views

How to prove this sequence is unbounded?

Let $a_{n} = \dfrac{7n^{3} - 3n^{4} -1}{4n^{2} + 3}$ How to show this sequence is unbounded without using limits? Well I know that I need to show that it unbounded from bottom or above. I choose ...
1
vote
3answers
47 views

Convergence test of the series [duplicate]

I need to prove that $$\sum_{n=1}^{\infty} {\sin{100n}} \; \text{diverges}$$ I think the best way to do it is to show that $\lim_{n\to \infty}{\sin{100n}}\not=0$. But how do I prove it?
2
votes
3answers
43 views

Proving $\int_{0}^{1}\sum_{n=1}^{\infty }\frac{4}{n }(\sin(\frac{2\pi nx}{3}))^3dx=\pi $

Proving $$\int_{0}^{1}\sum_{n=1}^{\infty }\frac{4}{n }\left (\sin\left (\frac{2\pi nx}{3}\right )\right )^3dx=\pi $$
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0answers
15 views

Show that for a sequence $p_{n}$ of real numbers, $\limsup p_{n} < +∞$ iff $p_{n}$ is bounded above.

Show that for a sequence ${p_{n}}$ of real numbers, $\limsup {p_{n}} < +∞$ iff ${p_{n}}$ is bounded above. My Partial Proof: $\Leftarrow$ Given ${p_{n}}$ is bounded above, prove $\limsup {p_{n}} ...
0
votes
2answers
27 views

$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$

I found in my archives solution of this exercise Calculate $$\sum\limits_{k=1}^{n^2}E(\sqrt{k})\quad n\in\mathbb{N}^{*}$$ E represent the floor function Solution: they made Let ...
1
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1answer
14 views

Interpret the following sequence $X_n^{(k)} = \underset{1 \leq i_1 < \dots <i_k \leq n }{\sum} \; \xi_{i_1} \dots \xi_{i_k}$

I'm working on a problem in which I have the following set up. Let $\xi_1, \xi_2, \dots$ be independent random variables with $E[\xi_i] =0$ for all $i$. Then they define the following sequence $ ...
3
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0answers
44 views

Nature of the series $\sum\limits_{n}(g_n/p_n)^\alpha$ with $(p_n)$ primes and $(g_n)$ prime gaps

Let $p_n$ denote the $n$th prime number and $g_n=p_{n+1}-p_n$ the $n$th prime number gap. This is to ask for which values of $\alpha$ the series $S_\alpha$ converges or diverges, where ...
-1
votes
1answer
21 views

Element-wise product of two series

Let $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ be two absolutely convergent series such that their values are $s_{1}$ and $s_{2}$ respectively. If $\sum_{n=1}^{\infty} a_{n}b_{n} = ...
1
vote
2answers
28 views

counter-example: aboslute convergence => convergence in incomplete vector space

Is the following statement true? Let $X$ be a normed linear space, $x_k \in X$, $k \in \mathbb{N}$ and $\sum_{k=0}^\infty \lVert x_k\rVert$ convergent. Then $\sum_{k=0}^\infty x_k$ is also ...
1
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2answers
52 views

Limit of $\frac{n!}{(n+1)!}$ as n approaches infinity.

I know that factorials grow faster than any exponential function, but what if you put two factorials up against each other? My problem is finding the limit of: $$\frac{n!}{(n+1)!}$$ as $n$ ...
1
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1answer
113 views

Is this infinite series related to prime and composite numbers convergent?

I don't know whether this series converges: $$\frac{1}{4} - \frac{1}{5} + \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} - \frac{3}{11} + \frac{1}{12} - \frac{1}{13} + ...
0
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0answers
15 views

Proof of convergence of Dirichlet's Eta Function

I'd like to check directly the convergence of Dirichlet's Eta Function, also known as the Alteranting Zeta Function or even Alternating Euler's Zeta Function: ...
1
vote
1answer
18 views

Is the series Convergence or divergence [on hold]

Examine the convergence of the series $$\sum_{n=1}^{\infty} \left(\sqrt[3]{n^3+1}-n\right)$$ Could you please help me to solve this?
0
votes
2answers
29 views

fibonnaci and lucas series technique

Well i have the following two problems involving fibonnaci sequences and lucas numbers, i know that they share the same technique, but i don't have clear the procedure: $$f_n = f_{n-1} + f_{n-2}: f_0 ...
1
vote
1answer
15 views

Convergence for x values of function

I'm trying to determine at what x values an infinite series of the function $-sin(nx)$ converges. I think I may be over thinking this relatively simple question. But I just want to verify that I'm on ...
0
votes
2answers
26 views

boundedness of the sequence $a_n=\frac{\sin (n)}{8+\sqrt{n}}$

How can i prove boundedness of the sequence $$a_n=\frac{\sin (n)}{8+\sqrt{n}}$$ without using its convergence to $0$? I know since it is convergent then it is bounded.
0
votes
1answer
22 views

Values of x for which series can converge

I'm given a function $sin(nx)/(n^2)$ and I'm trying to find for which values of x the infinite series for this function would converge. It's easy to see that $sin(nx)$ is always between (-1,1), so ...
0
votes
1answer
30 views

Finding Closed form of sum of Harmonic progression [on hold]

How to arrive at given closed form of following sequence? $$\left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}\right) + \left(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}\right) +\left( 1 ...
0
votes
1answer
17 views

Sequence of quadratic polynomials

Let $P_n$ be a sequence of Quadratic polynomials on $[0,1]$ such that $\lim_{n \rightarrow \infty}P_n(a_i) = b_i$ for $i = 1,2,3$ where $b_i$ are real numbers. Then 1) $P_n$ converges pointwise in ...
4
votes
1answer
61 views

Uncountable series without axiom of choice

Consider a sequence of positive real numbers $(\alpha_i)_{i\in I}$ for some (suppose maybe wellordered for now) set $I$. Using axiom of choice, it is easy to see that $\sum_i \alpha_i$ is infinite if ...
0
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0answers
15 views

Taylor Polynomials: Estimating accuracy of an approximation f(x) ≈ Tn(x)

$f(x) = \sqrt{x},\space\space\space\space a = 4,\space\space\space\space n = 2,\space\space\space\space 4 \le x \le 4.7$ I approximated f by a Taylor polynomial with degree 2 at the number 4. ...
0
votes
1answer
17 views

A basic logical question on on sequence of functions

Let $f_n:[0,\infty)$ and $f:[0,\infty)$ be a sequence of functions such that for every finite $T$, $f_n:[0,T]\rightarrow f:[0,T]$ uniformly. But, it need not be true that $f_n:[0,\infty)\rightarrow ...
-2
votes
2answers
65 views

How to differentiate a series? [on hold]

I need help with this function. Can this function be differentiated? $\frac{\partial}{\partial b}\ln{L}=\frac{n}{b}+\sum{\ln{x}}-\frac{\partial}{\partial b}(-a^b\sum{x^b})$ I dont know how to last ...
3
votes
2answers
33 views

Finding a generalized form for this series

While i was just playing around with series i came across this one, $$ S = \sum_{k=1}^\infty[\frac{k}{k-\frac{1}{2}}+\frac{k-\frac{1}{2}}{k}-\frac{k+\frac{1}{2}}{k} - \frac{k}{k-\frac{1}{2}}] $$ ...
1
vote
2answers
48 views

Proving Polynomial is Analytic

If a function $f$ at $x = a$ equals it's Taylor Series, $f$ is said to be analytic. So, if I were given a polynomial $p(x) = \sum_{n=0}^{200}{a_nx^n}$, and trying to prove that $p(x)$ was analytic ...
2
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1answer
33 views

Understanding a step in a double series proof

I'm really confused, how do they get from the first line to the second line ? $$\begin{align*} ...
1
vote
1answer
8 views

Show that $4 - Un+1 < 1/2(4 - Un)$

Let Un be a sequence such that : U0 = $0$ ; Un+1 = $sqrt(3Un + 4)$ We know (from a previous question) that Un is an increasing sequence and Un < $4$ Show that $4$ - Un+1 <(or =) 1/2(4-Un) I ...
3
votes
1answer
32 views

Absolute convergence of $ \sum a_nx_n $ implies absolute convergence of $ \sum a_n$

I'm trying to find a proof (or a conunter example, but I'm somehow convinced that the statement is true) for the following fact: $$ \forall_{(x_n)_{n=1}^{\infty} \lim{x_n} = 0 } ...
2
votes
1answer
59 views

Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series. $$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$ We know that if ...
2
votes
1answer
31 views

Series similar to harmonic function [on hold]

How to prove that series $\sum_{n=1}^{\infty}{(-1)^{\lfloor{\sqrt{n}\rfloor}}\frac{1}{n}}$ converges?
0
votes
2answers
23 views

What is the value of the following summation?

Compute $$\displaystyle\sum \limits_{n=0}^\infty (-1)^{n+1} \frac{1}{9^n(2n+2)}$$ I am given the fact that $$ \frac{1}{2}\ln(1+x^2) = \sum \limits_{n=0}^\infty (-1)^n\frac{x^{2n+2}}{2n+2} $$ ...
1
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0answers
30 views

Closed-form expressions of $\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$

Does anybody know if there's a closed-form expression of this series? $$\sum_{n=1}^\infty \frac{\sin^2(an) e^{-bn^2}}{n^2}$$ where $a$ and $b$ are strictly positive. It's easy to see that it's ...
1
vote
0answers
34 views

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$?

Is there a name for expressions of the form $\sum_{i=1}^N \prod_{k=1}^i a_k$? If so, where can I find the equivalent of a Wikipedia entry?
0
votes
1answer
24 views

Recurrence Relation, a question about the relation between $A_n$ and $A_{n+1}$

Given the following recurrence relation: $$A_{n+1} = {1 \over 4(1-A_n) }$$ $$ A_1 = 0 $$ Can I safely assume that if- $$\forall n \in \mathbb{N}, \ A_n < 1/2$$ then $$\forall n \in \mathbb{N}, \ ...
1
vote
4answers
89 views

Infinite sum of products on my mid-term

This problem was on my calculus mid-term : Determine the convergence or divergence of the series $$ \sum\limits_{n = 1}^\infty {\prod\limits_{k = 1}^n {\frac{{4k - 3}}{{4k - 1}}} } $$ I tried ...
3
votes
2answers
60 views

If $(1+x+x^2)^{25} =\sum^{50}_{r=0} a_r x^r$ then …

If $$(1+x+x^2)^{25} =\sum^{50}_{r=0} a_r x^r$$ then find : $\sum^{16}_{r=0} a_{3r} =$ My approach : let (1+x) =t therefore, $(1+x+x^2)^{25} =\sum^{50}_{r=0} a_r x^r$ =$(1+x+x^2)^{25} = ...
0
votes
0answers
16 views

Rational Binomial for Taylor series

I want to write this as a sum while $x_0 = 0$ $$f(x) = e^{-x}(1-x)^{-1/2}$$ I know the sum for $e^{-x}$ but I can't figure out the sum for $(1-x)^{-1/2}$ What I tried (minuses cancel out because of ...
0
votes
2answers
22 views

Prove that Recurvisv limits are equal [on hold]

Prove that: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} a_{n-1}$ Ideas?!?
3
votes
2answers
45 views

Uniform convergence of $\sum f(x)^n$

Let $f:X\to\mathbb{R}$ be such that $\sup\{|f(x)|:x\in X\}<1.$ Show that $\sum_{n=1}^{\infty} f(x)^n$ converges and compute the sum.. Every value given by $f$ is less than one, then if ...
1
vote
1answer
32 views

Convergence of the sequence $f_n(x)=\frac{1}{1+nx^2}$

I'm trying to find the convergence of $f_n$ and $f_n'$ where $f_n(x)=\frac{1}{1+nx^2}$. From the function if I derivate the result is $f'n= -\frac{2nx}{(1+nx^2)^2}$. To determine $f$ I have to take ...
1
vote
2answers
43 views

What is the limit of this sequence?

Problem 3 in the Exercises after Chapter 3 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $s_1 \colon= \sqrt{2}$, and let $$s_{n+1} \colon= \sqrt{2+\sqrt{s_n}} \mbox{ for ...
2
votes
1answer
19 views

Cauchy Product $n$ times

I'm looking for a short and precise proof of the following identity; $$\left(\sum_{k=0}^\infty \frac{C_k}{k!}x^k\right)^n=\sum_{k=0}^\infty\left[ ...
1
vote
1answer
19 views

What is the difference between arithmetic and geometrical series?

What is the difference between arithmetic and geometrical series? Also what are they? How do they look like?
3
votes
5answers
56 views

Taylor Series of $2xe^x$

I have to find the Taylor Series for $2xe^x$ centred at $x=1$. I came up with the following. $$e^x = e^{x-1} \times e = e \bigg( \sum_{n=0}^\infty \frac{(x-1)^n}{n!}\bigg)$$ Then consider $2xe^x$. ...