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

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3answers
27 views

Working on sequence, possibly recursive

I am working on this problem which asks to find if the sequence converges or not and if so the value it converges to. I am not sure how to deal with this type of question, but I feel like it may be a ...
0
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1answer
55 views

Let ${a_n},{b_n}>0 ,\lim \limits_{n \to \infty} [a_n+b_n]=0 $ then$ \lim \limits_{n \to \infty}a_n=0 $ and$ \lim \limits_{n \to \infty} b_n = 0$

Let ${a_n}$ and ${b_n}$ be sequences of nonnegative numbers. Show that if $\lim \limits_{n \to \infty} [a_n+b_n]=0$ then $\lim \limits_{n \to \infty}a_n=0$ and $\lim \limits_{n \to \infty} b_n = 0$. ...
1
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1answer
46 views

Find the function continuous or discontinuous

$\sum_{n=1}^∞ $ $(x+2)^n \over n! + x^2$ , Interval = [1,2] Is this function continuous in that interval ? I tried but the factorials are troubles.
5
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1answer
81 views

convergence of $\prod_{n=1}^\infty (1-\frac{z}{n!})$

I want to show that $\prod_{n=1}^\infty (1-\frac{z}{n!})$ is convergent (or uniformly convergent) (z is complex) Can I use the Theorem: The infinite product $\prod_{n=1}^{\infty} (1+a_n)$ converges ...
-2
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1answer
55 views

Are these “infinity” sequences true? [duplicate]

For $1\over 3$, you get $0.\overline3$, which is $0.33333...$. The threes go on forever. You can't ask "What happens if it ends in an eight?" because it simply doesn't end. For SSSSS..., what if it ...
6
votes
1answer
114 views

Integral Representation of the Zeta Function

How does one get from this $$\zeta(s)=\sum_{k=1}^{\infty}\frac1{k^s}$$ to the integral representation $$\zeta(s)=\frac1{\Gamma(s)}\int_{0}^\infty \frac{x^{s-1}}{e^x-1}dx$$ of the Riemann Zeta ...
1
vote
3answers
87 views

Evaluate to find the sum of an infinite series [duplicate]

$∑_{n=1}^\infty$ $n\over2^{n-1}$ or 1 + $2\over2$ + $3\over4$ + $4\over8$ + $5\over16$ + $\ldots$ How to go about evaluating the above, showing that it sums to 4?
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2answers
31 views

Definition of a point $x$ in a Riemann sum.

$$\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k) \Delta(x)$$ I am interested in what $x_k$ is. On stackexchange I have seen $x_k$ being defined as: $$x_k = \Delta(x)(k) + a$$ ...
2
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3answers
98 views

$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))$

how can i show this sequence $u_n$ is divergent: $$u_n=\exp( n\log n-(n+\epsilon)\log(n+\epsilon))\quad n\in \mathbb{N}^*;\quad \epsilon \in (0,1)$$ My attempts: \begin{align*} u_n&=\exp( n\log ...
1
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1answer
50 views

Finding the coefficients of $h(z)$ laurent series

Consider: $$h(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ Find the coefficients $a_n$ of the Laurent Series of $h(z)$ centered at $z=-2$ I got this from the approach here: Infinite sum complex analysis ...
0
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3answers
55 views

Finding $\lim_{n \to \infty}\frac{1}{\log(n+1)} = 0$

Can we use Sandwich theorem to show that $$\lim_{n \to \infty}\frac{1}{\log(n+1)} = 0$$ I am not getting the proper estimate. Please give a hint.
1
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0answers
24 views

Looking for sequences with certain conditions.

Are there sequences $a_n>0$, $p_n \subseteq (0,1)$ so that (1) $\lim_{n \to \infty} a_n = \infty$ (2) $\lim_{n \to \infty} p_n =0$ (3) $\lim_{n \to \infty} \frac{a_n}{n} = 0$ (4) $\lim_{n \to ...
2
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2answers
53 views

Convergence of a sequence given by recursive relation

Let $x_1=a>0$ and $x_{n+1}=x_{n}+\frac{1}{x_{n}};n>1$. Then does the sequence $(x_n)$ converge? The sequence $(x_n)$ is increasing. But I could not show that it is bounded. Any hint in this ...
3
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1answer
32 views

How to Derive this Digamma identity?

I dont see the transition from $(-z)^k$ in the fist sum to the transition to $(z+2)^k$ in the second sum? How is that derived?
1
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1answer
52 views

Find the residue at $z=-2$ for $g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$

Find the residue at $z=-2$ for $$g(z) = \frac{\psi(-z)}{(z+1)(z+2)^3}$$ I know that: $$\psi(z+1) = -\gamma - \sum_{k=1}^{\infty} (-1)^k\zeta(k+1)z^k$$ Let $z \to -1 - z$ to get: $$\psi(-z) = ...
2
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0answers
22 views

Residue of $f(z)$ using Laurent Series at $z=-2$ [duplicate]

Calculate the residue of: $$f(z) = \frac{\psi(-z)}{(z+1)(z+2)^3} \space \text{at} \space z=-2$$ Where $\psi(z)$ is the digamma function, and $\zeta(z)$ is the Riemann-zeta function (below). The ...
4
votes
3answers
109 views

To show sequence $a_{n+1}= \frac{a_n^2+1}{2 (a_n+1)}$ is convergent

Let $a_1=0$ and $$a_{n+1}= \dfrac{a_n^2+1}{2 (a_n+1)}$$ $\forall n> 1.$ Show that sequence $a_n$ convergent. I tried to prove $a_n$ is less than 1 by looking at few terms. But i failed to prove ...
6
votes
4answers
151 views

Limit $\lim_{n \rightarrow \infty} \frac{1-(1-1/n)^4}{1-(1-1/n)^3}$

Find $$\lim_{n \rightarrow \infty} \dfrac{1-\left(1-\dfrac{1}{n}\right)^4}{1-\left(1-\dfrac{1}{n}\right)^3}$$ I can't figure out why the limit is equal to $\dfrac{4}{3}$ because I take the limit of ...
4
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2answers
39 views

Limit of a sequence of a function

Let, $f_{n}(x)=\dfrac{1}{n} \sum_{k=0}^{n} \sqrt {k(n-k)}{n\choose k}x^{k}(1-x)^{n-k}$ for $x\in [0,1],n=0,1,2,...$ If $lim_{n\to \infty}f_{n}(x)=f(x)$ for $x\in [0,1]$, then the maximum value of ...
1
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2answers
101 views

Sum $\sum_{x=0}^{\infty} \frac{x}{2^x}$

Calculate $\sum\limits_{x=0}^{\infty} \dfrac{x}{2^x}$ So, this series converges by ratio test. How do I find the sum ? Any hints ?
0
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1answer
33 views

Computing $\mathrm{erfi}(\theta)\exp(-\theta^2)$:

I'm looking to compute $f(\theta):= \mathrm{erfi}(\theta)\exp(-\theta^2)$ as efficiently as possible, to double precision, with a fairly wide radius of converge. Computing $\mathrm{erfi}(\theta)$ ...
0
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2answers
51 views

Find the McLaurin series for $f(x)=x/(x+1)$

Can you help me find the McLaurin of $f(x)=x/(x+1)$ ? I am new to this mathematical chapter and already tried but I do not think my result is correct.
0
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3answers
46 views

Does the following matrixpowers converge?

Given a $3\times 3$ Matrix: $$A= \begin{bmatrix} -2 & -16.8 & -16.8& \\ -1 & -4.8 & -5.8 \\ 1.4 & 8 & 9\\ \end{bmatrix} $$ Determine whether the powers $A^n, ...
1
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0answers
46 views

Nature of the infinite differential sum operator?

Consider the operator $$ Hf = f + f' + f'' +\cdots = \sum_{i=0}^\infty \left[ \frac{d^i f}{dx^i}\right] $$ I am trying to determine what $ Hf $ is entirely in terms of $f$. I note the following ...
1
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1answer
71 views

Is my proof that if $x_n$ is a sequence such that $x_n \rightarrow +\infty$ then $(1+\frac{1}{x_n})^{x_n}\rightarrow e$ correct?

I'm trying to prove that if $x_n$ is a sequence such that $x_n \rightarrow +\infty$ then $(1+\frac{1}{x_n})^{x_n}\rightarrow e$. Here is what I've done so far: We know that $\{(1+\frac{1}{n})^n\}_{n ...
3
votes
4answers
117 views

How to see if $\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$ converges

I want to see if $$\sum_{n=1}^\infty\frac{\sqrt{n+1}-\sqrt{n}}{n}$$ converges or not. I start with $$α_{n}=\frac{\sqrt{n+1}-\sqrt{n}}{n}=$$ ...
0
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1answer
27 views

How to prove $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = (n+1)^{p+1}-1$?

This is the solution $ \sum_{k=1}^{p+1} \binom{p+1}{k}S_{n}^{p+1-k} = \sum_{k=1}^{p+1} \binom{p+1}{k} \sum_{l=1}^{n}l^{p+1-k} = \sum_{l=1}^{n}(l+1)^{p+1}-l^{p+1} = (n+1)^{p+1}-1$ ? With $S_{n}^{p} = ...
1
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2answers
59 views

Average of an Unbounded sequence of postive numbers.

Could someone check my proof. More of a sketch really. Can it happen that $s_n>0$ for all $n$ and that $\limsup s_n=\infty$, although $\lim\sigma_n=0$, where $\sigma_n={\sum_{k=1}^ns_n\over ...
0
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0answers
22 views

expansion in reciprocal terms

Given a function of the form $$T(x) = \frac{1}{(\sum_{i=1}^{\infty}c_i x^i)^n}$$ Is there ever a way to write it as the sum of reciprocal terms in $x$ .i.e a series of the form ...
3
votes
1answer
111 views

Solution to the Basel Problem in complex analysis pole issue.

Solve: $$\sum_{n=1}^{\infty} \frac{1}{z^2}$$ Before you mark as duplicate, I have a problem with only the consideration of the pole, please read carefully! Consider: $$f(z) = \frac{\pi \cot(\pi ...
4
votes
3answers
176 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3}$ using complex analysis

Evaluate: $$S = \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)^3} \space \text{using complex analysis}$$ This my question: we need to consider a $f(z)$ such that, $$\frac{1}{2\pi i} \cdot\oint_{C_N} f(z) ...
0
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2answers
137 views

Proving that the sequence $a_n=\frac{2}{2n^2-3}$ is null by the definition of a null sequence

I have the following question: Use the definition of a null sequence to prove that the sequence $\{a_n\}$ given by $$a_n = \frac{2}{2n^2-3}$$ is null. I am struggling to see why...
0
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1answer
33 views

Uniform convergence of series of arccosines

The series $$\sum_{n=1}^\infty\arccos{{n^4x^4}\over{1+n^4x^4}}$$ supposedly converges uniformly on any interval $I$ for which $0\notin\overline{I}$ while ...
0
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0answers
27 views

Does sequence of quantiles $x_{p_n}$ converge to endpoint $x_F$ of cdf F?

Let $X$ be a RV with continuous cdf $F$. Let $p_n$ be a sequence in $(0,1)$ with $\lim_{n \to \infty}p_n = 0$ and $x_{p_n}=\inf \{y \in \mathbb{R}:F(y) \ge 1-p_n\}$ the $p_n$-quantile of F. Define by ...
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2answers
43 views

Limit of a Sequence $\displaystyle\lim_{n\to\infty} \frac{(-3)^n+10}{2^n-1}$

Find the limit of the following sequence: $\displaystyle\lim_{n\to\infty} \frac{(-3)^n+10}{2^n-1}$ I solved it the following way, and was wondering if what I did is correct: ...
1
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2answers
40 views

Proof of the angle sum identity for cos using series

Let $sin(x)=\sum_{n=0}^{\infty}{(-1)^n\frac{x^{2n+1}}{(2n+1)!}}$ and $cos(x)=\sum_{n=0}^{\infty}{(-1)^n\frac{x^{2n}}{(2n)!}}$.How using above definition prove that ...
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0answers
18 views

What fuctions/algorithm was used ingeneration of these numbers?

I have 11 lottery tickets(used) and I have discovered that in each ticket, the 3rd digit's value is +1 of the value of the 6th digit.I have 11 tickets, each ticket is composed of 16 digits.Would ...
1
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1answer
40 views

Guidelines for choosing integrand to get a sum.

The idea was to find: $$\sum_{n=1}^{\infty} \frac{\coth(n\pi)}{n^3}$$ As you see in the solution, they conveniently choose a $f(z)$ they chose: $$f(z) = \frac{\pi \cot(\pi z)\coth(\pi z)}{z^3}$$ ...
0
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0answers
34 views

Why is taking the sum of residues required?

As you see in the solution, I am confused as to why the sum of residues is required. My question is the sum: $$(4)\cdot\sum_{n=1}^{N} \frac{\coth(\pi n)}{n^3}$$ Question #1: -Why is the ...
0
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1answer
50 views

Is there an explicit formula for this recursive series of matrices

I want to get an efficient way of computing $P_k$ from $P_0$ that satisfies the following recursion: $P_k = FP_{k-1}F^T+Q$ Where $P_k$, $F$ and $Q$ are matrices ($Q$ is diagonal, if it changes ...
0
votes
2answers
76 views

Convention for dummy variables [duplicate]

I have just been introduced to the topic of integration. My teacher said: $$\int_{a}^{b} f(x) dx$$ Is the same thing as: $$\int_{a}^{b} f(\alpha) d\alpha$$ I asked her why, and she said "it is a ...
2
votes
3answers
58 views

Convergence divergence of $ \sum a_n$ and $\sum \frac{a_n}{n}$

With nth term test $ \sum a_n $ diverges but what about $ \sum \frac{a_n}{n}$ can we use comparison test by taking auxillary series $\sum \frac{1}{n}$?
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5answers
3k views

Is it true that the Fibonacci sequence has the remainders when divided by 3 repeating?

About this Fibonacci sequence, is it true that the remainders when divided by three repeat along with the sequence like this: Fibonacci sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
1
vote
1answer
46 views

How prove there exsit $M$ such $a_{n}\le M$

Define sequence $a_{n}$ and such $a_{1}=a$ where $a$ is postive constant, and such $$(a_{n+1})^3=\sqrt{a_{n}}+a_{n}$$ show that: there exsit constant $M$,such for any postive integer $n$,have ...
1
vote
0answers
37 views

Strange sequence and limsup, liminf and limit points

Let $a_{n}$ be defined recursively as follows: $a_{n} = \begin{cases} 0 &\mbox {if } n = 1 \\ 1 &\mbox{if } n = 2 \\ \frac{1}{2}(a_{n-2} - a_{n-1}) &\mbox{if } a_{n-1} = 0 \\ 1 - ...
0
votes
2answers
123 views

How to prove that the $limsup$ of a sequence is equal to its greatest subsequential limit?

I have a very tricky problem that I'm having a hard time figuring out how to start. Basically, I want to prove that the supremum of the set of subsequential limits of a sequence is equal to the lim ...
3
votes
1answer
67 views

Limit of $nx_n$

Okay, so I have this problem: Given the sequence $(x_n)_{n\in\mathbb{N}}$ defined by $x_{n+1}=\dfrac{3x_n^2}{(1+x_n)^3-1}$, with $x_1>0$, find $\displaystyle\lim_{n\to\infty}x_n$ and ...
0
votes
2answers
169 views

Is the following sequence bounded below by zero?

The sequence $s_1=3$ and $s_{k+1}=\sqrt{3s_k-1}$, where $k$ is a natural number. My guess is that this is bounded below by zero. Is this correct? what is it bounded by? -- (Edit @Did: Here is the ...
1
vote
2answers
123 views

Is $\bigl(\sum {{x^n}\over{n!}} \bigr) \bigl(\sum {{y^n}\over{n!}} \bigr) = \bigl(\sum {{(x+y)^n}\over{n!}}\bigr)$ generalizable for series?

Before I had to do a proof demonstrating the properties of exponential multiplication using power series expansions: $$ e^xe^y=e^{x+y}, $$ and the easiest and quickest way I could think of doing this ...
4
votes
3answers
166 views

Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$

For the past week, I've been mulling over this Math.SE question. The question was just to prove convergence of an infinite sum, but amazingly WolframAlpha told me it had a remarkably simple closed ...