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

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0
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0answers
15 views

Multiplication of Limits when both diverges

I am working on the multiplication of limits, and I am able to prove when both converge, the multiplication converges to the multiplication of limits, however I cannot grasp the idea when for instance ...
0
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2answers
32 views

Let $a_n\in A$ and $\lim_{n \to \infty}|a_n-a_{n+1}|=0$,then which is true?

Let $a_n\in A$ and $\lim_{n \to \infty}|a_n-a_{n+1}|=0$,then which is true? $1.$There exists $a\in A$ such that $a=\lim_{n \to \infty}a_n$ $2.$There exists $b\in \mathbb{R}$ such that $b=\lim_{n \to ...
3
votes
1answer
48 views

Convergence of $\sum x_k \log (x_k^{-1}) $ [on hold]

Prove or disprove: If $\{x_k\} \subset (0,1)$, and $\sum x_k < \infty$, then $\sum x_k \log (x_k^{-1}) < \infty$. What if $\{x_k\}$ is monotone? Thanks a lot.
0
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0answers
18 views

Strong law of large numbers when sample size is a random variable

For a sequence $X_1, X_2, \ldots, X_n$ of i.i.d. random variables with mean $\mu$, the strong law of large numbers tells us that $$\sum_{i=1}^{n} \frac {X_i} {n} \xrightarrow{a.s.}\ \mu ...
1
vote
0answers
30 views

Question about series and how the pattern idea works

Two Questions: When you are given: $1, 2, 3, .... , n$ How do you know that in the $...$ that it continues the $x_{n-1} + 1$ pattern? Is it the definition of series? Secondly: Do partial sums ...
2
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0answers
35 views

Prove that $\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$

Put $g_n = 1 + \frac{1}{2} + ... + \frac{1}{n} - \log(n)$. Prove that $$\lim_{n\to\infty} \left( \frac{ g_n^{\gamma}}{\gamma^{g_n}} \right)^{2n} = \frac{e}{\gamma}$$ I've tried this for a while now ...
1
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0answers
35 views

Simple finite series with reciprocal factorials

I'm trying to find the following sum: $$ \sum_{k=0}^n{1\over(n-k)!}{x^{k+2}\over k+2}. $$ The most obvious way is to differentiate wrt to $x$ leading to $$ ...
1
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3answers
40 views

Is sequence limited and what is limit

I am stuck at one problem. So I have to check if sequence is convergent. $$\frac{2^x}{x!}$$ My thinking was to calculate limit and if limit exists it's convergent, but I am struggling with this: ...
3
votes
4answers
416 views

How to show this series diverges

How to show the following series $$\sum_{n=2}^\infty \frac{1}{n \log n}$$ diverge? Thank you!
0
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0answers
40 views

which of the following is sufficient for $\displaystyle \lim_{n\to \infty} \frac{a_n}{b_n}=1$?

Suppose sequences $a_n,b_n$ both have limits and are finite then which of the following is sufficient for $\lim_{n\to \infty} \frac{a_n}{b_n}=1$? $1.\lim_{n\to\infty} a_n=\lim_{n\to \infty} b_n $ ...
-1
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1answer
26 views

How to derive this inequality

I learnt that for a standard normal random varialbe $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
2
votes
1answer
30 views

The sequence $\sin \left({n\pi}\over 6\right)$ has the superior limit $L=1\dots$

I am studying limit points of a sequence now, and have some misunderstandings. Here's an exercises I have: The sequence $$\sin \left({n\pi}\over 6\right)$$ has the superior limit $L=1$and the inferior ...
5
votes
3answers
108 views

Is there any method to get a finite sum for $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots$?

As we can see on Wikipedia, there are some algebraic methods that give us finite sums for the Grandi's series $$1-1+1-1+1-1+1-1+\cdots$$ Let $S$ be the sum of the Grandi's series. Then ...
0
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2answers
40 views

Simple equivalent of the rest of the series $\sum\limits_n\frac1{n^3}$

Consider the converging series \begin{equation} \sum_{n\geqslant1}{\frac{1}{n^3}} \end{equation} I want to find an equivalent of the rest : \begin{equation} R_n=\sum_{k=n+1}^{\infty}{\frac{1}{k^3}} ...
2
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0answers
41 views

Time to buy a house without a mortgage equation!!

I am looking into a "real world" calculation to calculate the time taken for someone to buy their own home while they rent it. They do this by buying small pieces of the property every month, and ...
1
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1answer
30 views

Is there a way to simplify $\sum_{j=0}^{n}C_{n}^{j}\sum_{i=0}^{m}\frac{j}{j+i}C_{m}^{i}$

Is there a way to simplify the following sequence? $$ \sum_{j=0}^{n}C_{n}^{j}\sum_{i=0}^{m}\frac{j}{j+i}C_{m}^{i} $$
-4
votes
0answers
19 views

How to add these two sequences?

What would the answer be if you added these two Fibonacci sequences together? F_[m+k] = F_[m-1] F_[k] + F_[m] F_[k+1] F_[m+k-1] = F_[m-1] F_[k-1] + F_[m] F_[k]
4
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1answer
34 views

Sequences and limits

Let (an) be a sequence with an>0 for all natural numbers n. Assume that lim(an)=0. Show that the set of all numbers an has a maximum. That is, show that there is some number p, such that an <=ap. ...
2
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3answers
78 views

If a set $S\subset\mathbb R$ is not closed, does it contain a convergent sequence with a limit outside of $S$?

Suppose S is a subset of R and that S is not closed. Must it follow that there is a convergent sequence in S that converges to some l not in S?
5
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2answers
101 views

Evaluating a series of hypergeometric functions

I would like to prove (or disprove) the following statement: $$ \sum_{n=0}^\infty \left[\frac{{}_2{\rm F}_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3}{2};1\right)}{n!}\right] = \frac{\pi}{2} \left[ ...
1
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2answers
42 views

Determine whether or not $\sum_{k=1}^{\infty} \frac{1}{k- \mathrm{e}^{-k}}$ converges.

I have the following so far Let $a_k = \frac{1}{k- e^{-k}}$. Now, $\lim_{k \to \infty}a_k=0 \implies$ $\sum a_k$ can either converge or diverege. We must thus do further tests to determine whether ...
0
votes
1answer
51 views

Why ${1\over n+1}+ \cdot\cdot\cdot +{1\over 2n}>{n\over 2n} $?

Can you please explain me why is : ${1\over n+1}+ \cdot\cdot\cdot +{1\over 2n}>{n\over 2n} ={1\over 2}$? Thank you very much.
0
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3answers
17 views

Expanding the series …

Here we have such a sequence $x_n$. The thing that I do not understand is the following: where does the right side of this equality come from, how is it formed ? Can you please show me the operation ...
2
votes
1answer
44 views

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge

Suppose the limit of $f(z)$ as $z$ approaches $z_0$, exists and call it $w_0$. Suppose a sequence $(a_n)$ converges to $z_0$. Does $f(a_n)$ converge to $w_0$ and $n \rightarrow \infty$? I would say ...
2
votes
2answers
16 views

In $(]0,1], d_{\mid.\mid})$ why $(\frac{1}{1+n})_{n\in \mathbb{N}}$ do not converge?

Can someone tell me why this sequence do not converge ? First, I know that is a Cauchy's sequence. Then, the fact is that the sequence converges to $0$ when $n \rightarrow \infty$. Thanks in ...
0
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0answers
12 views

Can Banach fixed-point theorem be generalized to the case where each term depends on multiple previous terms in recurrence sequence

Banach fixed-point theorem http://en.wikipedia.org/wiki/Banach_fixed-point_theorem I'm wondering if the theorem still applies if the recurrence relation is, for example, ...
1
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2answers
45 views

Accumulation points and closed sets

Denote by $F$ the set of all accumulation points of $(x_{n})$. We define an accumulation point $x \in \mathbb{R}$ if there exists a subsequence $(x_{n_{k}})$ of $(x_{n})$ (being the latter a bounded ...
0
votes
1answer
28 views

Subsequence converging to inf of sup

Let $(x_n)$ be a bounded sequence, and for each $n\in\mathbb{N}$ let $s_n=\sup\{x_k:k\geq n\}$ and $S=\inf\{s_n\}$. I need to show that there exists a subsequence of $(x_n)$ that converges to $S$. I ...
0
votes
2answers
32 views

Write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form $a+bi$

I have to write $\sum_{k=0}^{\infty} \cos(k \pi / 6)$ in form: $a+bi$. I think I should consider $z=\cos(k \pi / 6)+i\sin(k \pi / 6)$ and also use the fact that $\sum ...
0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
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2answers
43 views

Find the limit of a sequence $(\frac{z^n}{n!})_{n=1}^{\infty}$

I have to find the limit of a sequence $(\frac{z^n}{n!})_{n=1}^{\infty}$ where $z$ is a complex number. I think it is zero, because we know that $\sum_{n=0}^{\infty} \frac{z^n}{n!}$ is finite. Is this ...
-5
votes
3answers
41 views

Non-Existence of Geometric Series Going to Infinity [closed]

According to this material, when analyzing the series $S = 1 + 2 + 4 + 8 + ...$ $S_n = \sum_{k=0}^n 2^k$ does not approach a specific value, so we say that the sum of the infinite geometric series ...
1
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0answers
63 views
+50

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=\frac{(\sqrt3+2)^{2^{p-1}}+1}{(2^{p}-1)(\sqrt 3+2)^{2^{p-2}}}$$ is a natural number then $2^{p}-1$ is a prime ...
1
vote
1answer
19 views

What will be nth term of the following sequence?

Let a, a+d, a+2d,...., be an A.P.If we eliminate every pth term, then what will be the new general value of nth term? For e.g. Let the A.P. be 2,5,8, 11 ,14,17,20, 23 ,26,29...[a=2, d=3] Now, if we ...
1
vote
6answers
221 views

How to prove a sequence does not converge?

I want to show that the sequence $$a_n=\frac{1}{n}+(-1)^n$$ does not converge to a limit. I know that if a sequence $\left( a_n\right)_{n \in \mathbb{N}}$ converges to a limit L, then ...
-2
votes
1answer
42 views

When I have proven $\lim (S_n)=s$, where $S_n$ is a series. Am I allowed to say $\lim (S_n) = \lim (S_{n+1}) = \lim (S_{n+2}) = s$?

I don't know how to put low letters in the title, but I hope the question is clear.
0
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1answer
29 views

Discuss convergence and find sum of the Series

Show that the series $\sum_{n=1}^\infty \ln(1-\frac{{1}}{10^n})$ converges and find the sum in closed form if it is possible. Try:Clearly given series converges because if $0<a_n<1$ then ...
2
votes
3answers
55 views

Show that $\sum_{k=1}^\infty \frac{\ln{k}}{k^2}$ converges or diverges

Show that $\sum_{k=1}^\infty \frac{\ln{k}}{k^2}$ converges or diverges I know that I must use the Limit Comparison test and my instinct tells me that this series will converge. I cannot, however, ...
0
votes
0answers
26 views

Analysis .Find sequences? [closed]

Find sequences $a_n, b_n$ such that \begin{align*} &a_n=\frac{x_n}{y_n} 3^{1/2}, \quad a_n \rightarrow 1 \\ &b_n = \frac{z_n}{w_n}, \quad b_n \rightarrow 3^{\frac{1}{2}} \end{align*} where ...
4
votes
8answers
117 views

Calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$

How can I calculate $1+(1+2)+(1+2+3)+\cdots+(1+2+3+\cdots+n)$? I know that $1+2+\cdots+n=\dfrac{n+1}{2}\dot\ n$. But what should I do next?
1
vote
1answer
160 views

Prove $\sum_{n=1}^\infty \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}$ converges and evaluate it.

Noting that the series inside is a part of the alternating harmonic series multiplied by $-n$, we get $\displaystyle \sum_{m=2^{n+1}} ^\infty \frac{(-1)^mn}{m}=\sum_{m=1} ^\infty \frac{(-1)^mn}{m} ...
1
vote
1answer
38 views

Evaluating $ \sum_{i=0}^{\infty}ir^i$ for $|r|\lt 1$ [duplicate]

I'm doing my mathematics homework and there's question which I'm pretty much unable to solve. I've literally tried every method but no results. It'd be great, if anybody could help me. Thanks (in ...
0
votes
3answers
55 views

finiding $a_n$ if $a_1=2,\; a_2=3,\; a_{n+1}=3a_n-2a_{n-1}$

Given $a_1=2,\; a_2=3,\; a_{n+1}=3a_n-2a_{n-1}$. How can i prove that: $a_n=2^{n-1}+1$ I Tried to isolate $a_n$ but it doesn't get me anywhere. Thanks.
2
votes
0answers
39 views

How to calculate alternating Euler sum [closed]

In How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$? get $$\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2=\frac{1}{12}(\pi^2\log2-4(\log 2)^3-9\zeta(3)),$$ Similar, how to evaluate the series, ...
0
votes
0answers
41 views

Prove probing to be permutation

So I have been taught that probe sequences (h(k, 0), h(k, 1), ... , h(k,m - 1)) are meant to be a permutation (0, 1, ... ,m - 1), but how does one prove that? I was asked this question in an ...
5
votes
3answers
127 views

Some integral representations of the Euler–Mascheroni constant

What kind of substitution should I use to obtain the following integrals? $$\begin{align} \int_0^1 \ln \ln \left(\frac{1}{x}\right)\,dx &=\int_0^\infty e^{-x} \ln x\,dx\tag1\\ &=\int_0^\infty ...
0
votes
2answers
40 views

How to decide about the convergence of $\sum(n\log n\log\log n)^{-1}$?

In Baby Rudin, Theorem 3.27 on page 61 reads the following: Suppose $a_1 \geq a_2 \geq a_3 \geq \cdots \geq 0$. Then the seires $\sum_{n=1}^\infty a_n$ converges if and only if the series $$ ...
2
votes
2answers
46 views

Proving $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula

I am trying to show that $z\,\Gamma(z) = \Gamma(z+1)$ using the product formula: $$ \Gamma(z) = \frac{e^{-\gamma z}}{z} \prod\limits_{n=1}^\infty\left(1+\frac{z}{n}\right)^{-1}e^{z/n}$$ where $\gamma$ ...
2
votes
1answer
29 views

How should Theorem 3.22 in Baby Rudin be modified so as to yield Theorem 3.23 as a special case?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, 3rd edition, and am at Theorem 3.22. Theorem 3.22: Let $\{a_n\}$ be a sequence of complex numbers. Then $\sum a_n$ converges if and ...
1
vote
1answer
18 views

convergence criteria of an infinite series

$\sum _{n=1}^{\infty }{\frac {1}{50}}\,{\frac { \left( -1 \right) ^{1+n }{\it a}\, \left( 10000\,\cos \left( tn \right) \epsilon\,\delta_{{ 1}}-10000\,\cos \left( \frac{1}{10}\,\sqrt {4201}t \right) ...