Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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43 views

I have written a maths “paper” about a nice trick that I discovered. Does anyone mind reading it and giving me feedback.

file:///C:/Users/attar/OneDrive/Desktop/The%20Hidden%20Sequence.pdf This is the link to the pdf. To put it into some context: I am a 16 year old mathematics student who is passionate about the subject....
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4answers
53 views

$\sum\left(\frac{ n^2 + 1}{n^2 +n + 1}\right)^{n^2} $ converges or diverges?

The original question is to show that $\;\sum\left(\dfrac{ n^2 + 1}{n^2 +n + 1}\right)^{n^2} $ either converges or diverges. I know it diverges but I'm having difficulty arriving at something useful ...
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2answers
29 views

Infinity minus a divergent series

I was testing out Wolfram Alpha’s language to math feature and input this query: My understanding is that this would be undefined, as the series diverges; however wolfram states the answer is ...
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0answers
11 views

Using formal definition of a limit to prove (-1/2)^n converges to L = 0

Hey there I'm trying to solve a convergence question that uses the formal definition of convergence to show that (-1/2)^n converges to L = 0. I have gotten up to the below stages but I am unsure if it ...
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0answers
26 views

Proving two polynomials converge to the same function

I am looking to create polynomials that converge past the usual radius of convergence on the real line. So far, I have proved that this polynomial will converge to the analytic continuation of the ...
2
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2answers
70 views

Convergence of $a_1 =1$ and $a_{n+1} = 1 + \frac{1}{1+a_n}$.

Let the sequence $(a_n)$ be defined as follows: $a_1 =1$ and $a_{n+1} = 1 + \frac{1}{1+a_n}$. To prove the sequence converge and find the limit. I have observed the first few terms of the sequence: $...
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4answers
32 views

Why is it that $(n+(n+1)^2+(n+2)^3) \bmod 144 =0$ starting with $n=3$ and incrementing $n=n+6$?

The series involves: $(n+(n+1)^2+(n+2)^3) \bmod 144 =0$ starting with $n=3$ and incrementing $n=n+6$ Examples: $3+4^2+5^3 = 3+16+125 =144$ and $144 / 144 =1$ $9+10^2+11^3 = 9+100+1331 =1440$ and $1440 ...
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1answer
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A sum involving 3 constants

How may one show that: $$\sum_{n=0}^{\infty}\frac{{2n \choose n}^3}{2^{6n}}\frac{n}{(2n-1)^4}\sum_{k=0}^{n}(-1)^k{n \choose k}\frac{(n-k)[2\phi^4k^2+k+\phi^2]}{2k-1}=-\frac{\phi^2 G}{\pi}$$ Where: $G=...
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1answer
23 views

Coefficient in Formal Power Seies

Find the coefficient of $z^k$ in $$\frac{\sum_{n\geq1} z^n/n}{1-z}$$ Is my solution correct? Write the above sum as: $$\sum_{n\geq1} z^n/n\cdot\sum_{n\geq0} z^n$$ To get the coefficient of $z^k$ we ...
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If a sequence converges, prove a sequence of partial sum also converges. [duplicate]

If the sequence $\{a_{n}\}$ of real numbers converges to $a \in \mathbb{R}$, prove that the sequence $\{b_{n}\}$ where $b_{n} = \frac{1}{n} \sum_{k=1}^{n} a_{k}$ also converges to $a$. Prove also that ...
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2answers
43 views

Find the power series for $\ln(9-4x^2)$

I am trying to get the power series for $\ln(9-4x^2)$ The first step I took was taking the derivative of $\ln(9-4x^2)$ which I got: $\frac{-8x}{9-4x^2}$ Getting this into the form of $\sum_{n=1}^\...
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4answers
69 views

How can I prove that $\sum_{n=1}^\infty \dfrac{n^4}{2^n} = 150 $?

I can easily prove that this series converges, but I can't imagine a way to prove the statement above. I tried with the techniques for finding the sum of $\sum_{n=1}^\infty \frac{n}{2^n}$, but I didn'...
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1answer
35 views

Taylor Series representation of $f(x) = \sqrt{x} + \frac{1}{\sqrt x}$ at $a=1$

I am trying to find the Taylor Series representation of $f(x)= \sqrt x + \frac1{\sqrt x}$ at $a = 1.$ With $5$ terms. I know how to get the series expansion. centered at $a=1$. with $5$ terms… However ...
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35 views

Equivalent for $v_{n+1}=v_n\ln(v_n)$

Do you know how to get an equivalent for $(v_n)$ defined by $v_0>\mathrm{e}$ and $\forall n\in\mathbb{N},\; v_{n+1}=v_n\ln(v_n)$ ? Thank you.
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Examine $|x_n|$ of a Sequence; 3 cases

Let $z=\sqrt(3)\frac{5}{12}+i\frac{5}{12}$ and let $b\in \mathbb{R}$. Define $x_n=(b\cdot z)^n$. Now, I know that if $|bz|<1$, then $x_n \to 0$ if $|bz|>1$, then $|x_n| \to \infty.$ if $|bz|=...
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1answer
32 views

Uniform convergence of series of functions. [closed]

Consider the sequence of functions $\left(f_n\right)_{n\in \mathbb{N}}$ where $$f_n(x)=\ln\left(\frac{x}{n} +1\right),\qquad n\in \mathbb{N}.$$ How do I prove that this sequence of functions does not ...
2
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1answer
32 views

Infinite Series: Convergence and Divergence tests

Good Day! I have come across a situation in an exercise which I am unable to figure out. I am sharing the details of the problem are stated below. The infinite series $\sum_{n=3}^{\infty} \frac{1}{n(...
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3answers
31 views

How does: $p\sum_{n=1}^{\infty}(1-p)^{n-1}$ simplify to $\frac{p}{1-(1-p)}$

I'm trying to figure out how the geometric random variable can be simplified to the RHS, and would appreciate some help in figuring this out! $p\sum_{n=1}^{\infty}(1-p)^{n-1}$ simplify to $\frac{p}{1-(...
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0answers
25 views

Is my reasoning correct to prove normal convergence of $\sum_{k}^{\infty}\ \frac{1}{n}(\frac{1}{2}\sin(x))^{n}$

So I have to prove/test every type of convergence (point wise, uniform, absolute, normal) of the following: $\sum_{k}^{\infty}\ \frac{1}{n}(\frac{1}{2}\sin(x))^{n}$ So I decided to start it of by ...
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1answer
54 views

Can the sum of all elements of a continuous, finite set of real numbers be expressed as an infinite sum of distinct numbers?

I'm a high-school math student, and I recently stumbled upon an interesting, but counterintuitive result while solving a problem. I was trying to prove that the sum $S$ of all $x$ such that $x\in(0,1)$...
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73 views

Does this series converge? By squeeze theorem.

Positive series $\{x_n\}$ satisfies $$\ln \frac{{{x}_{n+1}}}{{{x}_{n}}}=\frac{{{x}_{n+1}}-{{x}_{n}}}{{{x}_{n+1}}}+\frac{{{({{x}_{n+1}}-{{x}_{n}})}^{2}}}{2{{a}^{2}}},\;\; n\in\mathbb{N}^*$$ for ${{x}_{...
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Prove that the sequence $S_n=\sum_{i=n}^{\infty} x_i$ is cauchy?

Let the sequence $(x_n)_{n\in \mathbb{N}}$ be defined by $lim_{n\to \infty} x_n =0$ and $(x_n)_{n\in \mathbb{N}}$ is monotonically increasing. Prove that $S_n=\sum_{i=n}^{\infty} |x_i|$ is cauchy?
3
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1answer
51 views

If $a_n \neq 0$ for all natural numbers n , $\sum_{n=1}^{+\infty} a_n $ converges $\lim_{n\to+\infty} a_n/b_n =1\sum_{n=1}^{+\infty} b_n$ converges

If $a_n \neq 0$ for all natural numbers $n$ , $\sum_{n=1}^{+\infty} a_n $ converges $\lim_{n\to+\infty} \frac{a_n}{b_n} =1 $ then $\sum_{n=1}^{+\infty} b_n$ converges I am trying to find ...
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0answers
52 views

Prove that if $\sqrt[n]{x_n} \to \ell$ then $x_{n+1}/x_n \to \ell$

I am stuck with the following problem: Let $\lbrace x_n \rbrace$ be a sequence of positive real numbers. Prove that if \begin{align} \lim_{n\to \infty} \sqrt[n]{x_n} = \ell \Longrightarrow \lim_{n\...
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0answers
30 views

Convergence limit of a factorial sequence. [closed]

Show that the sequence $$ a_{n}=\frac{\pi}{2} \frac{((2 n) !)^{2}(2 n+1)}{2^{2 n}(n !)^{4}} $$ converges to 1 and conclude that $$ \lim _{n \rightarrow \infty} \frac{2^{2 n}(n !)^{4}}{(2 n+1)((2 n) !)^...
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2answers
57 views

Does the following converge to 0?

Does the following converge to $0$ for $\theta>-1/2$? $$\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^ni^{3\theta}}{\left(\sum_{i=1}^ni^{2\theta}\right)^{3/2}}$$ I'd like to use the comparison test but ...
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2answers
35 views

Convergance using Cauchy Criterion

let $$\sum_{k=1}^n \frac{1}{n+k}$$ The sum is convergent since it satisfies the Cauchy criterion: $$|z_{n+p} - z_n| = \left|\sum_{k=n+1}^{n+p} \frac{1}{n+k}\right| \le \sum_{k=n+1}^{n+p}\frac{1}{n+n+1}...
2
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2answers
46 views

Argue that the sequence converges

Let $$a_n=n^2\cos(1/n)-n^2$$ Show that the sequence converges. Now, I know how to use the formal definition of convergence but I am looking for simpler methods (i.e the tests for series). I found it ...
2
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2answers
128 views

Compute $\int_{0}^{1}\frac{\ln(1+x)\ln(1+x^2)}{x}\,dx=\frac{\pi}{2}G-\frac{33}{32}\zeta(3)$

I saw the following result and I am trying to prove it. $G$ is Catalan´s constant. $$\boxed{\int_{0}^{1}\frac{\ln(1+x)\ln(1+x^2)}{x}\,dx=\frac{\pi}{2}G-\frac{33}{32}\zeta(3)}$$ I could not figure out ...
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39 views

Spivak calculus 7. ch 22: Convergence of a continued fraction sequence

This exercise is has a "small" modification, but I haven't been able to solve it yet and I wanted some advice! thanks! For any positive integers $a$ and $b$ consider the sequence $$ a_{1}=a, ...
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2answers
41 views

Series that does not converge

Let $$\sum_{n=0}^{+\infty}\frac{x^n}{n+1}$$ with $x\in\mathbb{R}$. If $x<-1$, then I can write $$\sum_{n=0}^{+\infty}\frac{x^n}{n+1}=\sum_{n=0}^{+\infty}\frac{(-1)^n|x|^n}{n+1}$$ because $x=-|x|$, ...
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0answers
13 views

How do I prove a sequence is disjunctive?

I wrote a random number generator with an unbounded state size. I don't know where to begin proving it to be (or proving it isn't) disjunctive. What would be a property of a disjunctive sequence, ...
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1answer
24 views

Prove that the sequence of partial sum does not admit limit

I want to prove that given the series $$\sum_{n=1}^{\infty}(-1)^n a_n$$with $a_n\geq 0$ and increasing. Now then I want to prove that if I consider $S_n=\sum_{k=1}^{n}(-1)^k a_k$, supposing that $S_n$ ...
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0answers
46 views

Lexicographically earliest sequence with shift-sum property

I use $\mathbb{N}$ to denote the set of non-negative integers. Let $a: \mathbb{N} \rightarrow \mathbb{N}$ satisfy $a(n+a(n+1)) = a(n) + a(n+1)$. Two trivial examples are $a(n) = 0$ and $a(n) = n$. But ...
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13 views

Is function a Markov chain?

Suppose we have a sequence of integer random variable $(X_k)_{k=0}^\infty$ which is Markov chain and function $f: \mathbb{Z} \rightarrow \mathbb{Z}$. That said, Is it true that function with sequence ...
1
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1answer
27 views

$\{a_n\},\{b_n\}\subseteq\mathbb{C}$. If $\sum_{n=1}^\infty a_n(b_n-b_{n+1})$ and $\{a_n b_{n+1}\}$ converges, $\sum_{n=1}^\infty a_nb_n$ converges.

I tried to use $$\displaystyle \sum_{n=M}^N a_nb_n=b_NA_N-b_NA_{m-1}-\displaystyle \sum_{n=M}^{N-1}(b_{n+1}-b_n)A_n, where A_k=\displaystyle \sum_{n=1}^ka_n, and A_0=0$$. But I get thinks like $$\...
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0answers
28 views

Double Summation index change

The generating function of the Bessel function can be written as: $$g_{(x,t)} =\sum_{r=0}^{\infty}\sum_{s=0}^{\infty} (-1)^s (\frac{x} {2})^{r+s} \frac{t^{r-s}} {r!s!} $$ Changing the summation index $...
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0answers
44 views

Closed expression for $\sum_{k=0}^{N-1}k^2$?

I have to perform the following power sum in my embedded device $$\sum_{n=0}^{N-1}({x[n]})^2 \tag1$$ with $x[n] = 0,1,2,3...(N-1)$ Hence, I think that I can rewrite above power sum as $$\sum_{k=0}^{N-...
3
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2answers
39 views

Limit of the sequence $x_{n+1}=x_n(2-ax_n)$ For some real $a$ positive.

Find the limit of the following recurrence relation $$x_{n+1}=x_n(2-ax_n)$$ For some real $a$ positive. I don't know how to find a closed form of the given recurrence relation and since I don't have ...
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2answers
34 views

Finding values where a series converges

So I was given the following prompt: "Determine the values for which the series converges" $\sum_{n=1}^\infty(-1)^n(n!)(x-3)^n$ I guess I'm a bit confused over where I'd start here, I ...
1
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2answers
43 views

Find explicit formula for summation for $ p > 0$

Find explicit formula for summation for any p>0: $$\sum_{k=0}^n\frac{1}{(k+1)p^k}\binom{n}{k}$$ I know that $$(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k$$ but still have no ideas how to do this? Can ...
1
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1answer
21 views

Find the order of convergence of the given sequences

To solve $x^2+x-1=0$ we have used the two following sequences: $$x_{n+1}=\frac{1}{x_{n}+1}\;\;\;,\;\;\;x_{n+1}=\frac{x_{n}^{2}+1}{2x_{n}+1}$$ Find the order of convergence of both of them. The ...
2
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1answer
21 views

I need help with this question about convergent series

Let $\sum_{n=1}^{∞} a_n$ be a convergent series, consider a growing sequence where $n_1<n_2<...$ , and define • $b_1=a_1+...+a_{n_1}$; • $b_2=a_{n_1+1}+...+a_{n_2}$; • $b_3=a_{n_2+1}+...+a_{n_3}$...
2
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2answers
57 views

Show that $\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n}(-1)^kf\left ( \frac{k}{n} \right )=0$

Let $f : [0, 1] → \mathbb{R}$ be a continuous function. Prove that $$\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{k=1}^{n}(-1)^kf\left ( \frac{k}{n} \right )=0$$ First, I observed that any pair of ...
3
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1answer
68 views

Why $\sum _ {n = x}^{\infty} (-1)^n\left (n^{1/n} - 1 \right) $ gives so many repeating 0 s here?

$\sum _ {n = 1}^{\infty} (-1)^n\left (n^{1/n} - 1 \right) $ is the MRB constant. Whether it's rational or not I know its terms and all of its partial sums are irrational. So why does $\sum _ {n = x}^{\...
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0answers
25 views

Is there an iff Theorem to say the series of function uniformly converges?

Let $(f_n)$ be a sequence of real-valued function defined on a set $X$ . Suppose that $f_n(x) ≥ 0$ for any $x ∈ X$ and any $n ∈ \mathbb{N}$ $f_n(x) ≥ f_{n+1}(x$) for any $x ∈ X$ and any $n ∈ \mathbb{...
0
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1answer
49 views

$\sum_{n=1}^{\infty}(\frac{3n}{3n-1})^{n!}$

Regarding this sum $\sum_{n=1}^{\infty}(\frac{3n}{3n-1})^{n!}$ I need to tell if it converges or diverges. Here is my attempt: $\sum_{n=1}^{\infty}(\frac{3n}{3n-1})^{n!} = \sum_{n=1}^{\infty}(\frac{3n-...
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0answers
13 views

Briefly explain how you conduct the following Econometrics tests: [closed]

i. Multi- collinearity ii. Heteroscedasticity iii. Serial correlation
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27 views

Can you use substitutions to evaluate summations?

Suppose we have the sum $$\sum_{n=a}^{b}f(g(x))$$ Where $$g(a)<g(b)$$ and both are integers. Are we allowed to use the substitution $$m=g(n)$$ and get $$\sum_{m=g(a)}^{g(b)}f(m)$$ that gives the ...

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