For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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1answer
47 views

Are these recursive sequences convergent?

Fix an integer $k > 1$. Suppose $a_1,\ldots,a_k > 0$ and for $n > k$ we define $$a_n = 1/a_{n-1} + 1/a_{n-2} + \ldots + 1/a_{n-k}$$ Are these recursive sequences always convergent for any $...
2
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2answers
98 views

Finding the interval of convergence of $\sum^{\infty}_{n=0}\frac{(2n)!}{(n!)^2}x^n$

This is part of a question: Use a similar trick to find nice upper and lower bounds for $\frac {(2n)!}{4^n(n!)^2}$, and thus finish finding the interval of convergence of $\sum^{\infty}_{n=0}\...
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3answers
736 views

different types of number series and their uses in real life

What are the different types of number series that have practical uses in real life. e.g. Fibonacci series, it is very important series which is present and useful in our nature. Please can you ...
0
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1answer
34 views

Modular inequality of sequential terms: $|x_ny_n-xy| \le |x||y_n-y|+|y_n||x_n-x|$

How can I prove that $|x_ny_n-xy| \leq |x||y_n-y|+|y_n||x_n-x| $ ?
0
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2answers
45 views

Does the series converge or diverge and how can you tell

The series I am stuck on is $$\sum_{n=3}^\infty\frac{\sqrt n}{2n-1}$$ I am not sure howto tell whether it converges or diverges. I tried the ratio test and i get infinity/infinity. When I graph ...
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3answers
296 views

Absolute convergence to zero imply convergence to zero?

Given that $\frac{1}{N}\sum_{i=1}^N {|a_i|}$ converges to zero as $N\rightarrow \infty$, does it imply that $\frac{1}{N}\sum_{i=1}^N {a_i}\rightarrow 0$? I know absolute convergence imply convergence,...
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1answer
26 views

Finding covergence/divergence of series using the integral test

I have the series: $\sum_{n=1}^\infty (1+{1 \over n})^n$ And I need to find if it converges or diverges using the integral test. I think that the sequence of $(1+{1 \over n})^n$ Will converge to ...
1
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1answer
22 views

Appell Sequences as Group under Addition

I was wondering if infinite polynomials sequences such as Appell Sequences could be groups under addition. Take a polynomial sequence $\{p_n(x)\}$ where the following condition is imposed: $$p'_n(x)=...
2
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1answer
79 views

Rearranging $\sum_{k = 0}^{+\infty} \left(z+\frac{1}{2}\right)^k$.

Consider the complex series: $$\sum_{k = 0}^{+\infty} \left(z+\frac{1}{2}\right)^k.$$ Clearly the series converges for $\left|z+\frac{1}{2}\right| < 1$, by the ratio test. I am supposed to write ...
0
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2answers
169 views

Show that a Cauchy sequence has a fast-Cauchy subsequence

A sequence $\{x_j\}$ is said to be fast-Cauchy if $\sum_1^\infty d(x_j,x_{j+1})<+\infty$. Show that every Cauchy sequence has a fast-Cauchy subsequence. **My attempt:**Argue by contradiction, ...
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1answer
33 views

Does the power series converge or diverge

The series is (pi/4)^k. I already know the series converges because it is a power series with r being less than one. The problem is, I am not sure how to test it. My first idea was to use the ...
1
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2answers
21 views

Conditionally convergent sequences and implications

If I have $\sum b_n$ is conditionally convergent, how can I show that $\sum b_{4n}$ doesn't in general converge? Assume $(b_n)$ is an arbitrary sequence of the Reals All I need is a counter example ...
1
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2answers
25 views

How to find convergence or divergence of a series?

How would I show that this is either convergent or divergent? $\sum_{n=1}^\infty {5^n \over 4^n + 3}$ I think that it is a geometric series, and that I should reformat it so that it is in the form $...
1
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1answer
46 views

Convergence behavior of $\sum_p \frac{1}{p \log p}$ and generalization.

The harmonic series $$\sum_{n\in\mathbb N} \frac{1}{n}$$ is well known to be divergent. Using Cauchy condensation test one immediately sees that even $$\sum_{n\in\mathbb N} \frac{1}{n\log n}$$ is ...
1
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2answers
57 views

What will be the sum of the series of binomial co-efficients?

What will be the sum of the following binomial co-efficent series $$\binom{z+1}{z} + \binom{z+2}{z} + \binom{z+3}{z} + \dots + \binom{z+r}{z} = \sum\limits_{i=1}^r \binom{z+i}{z}$$ Thank you
5
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1answer
87 views

Evaluation of the series $ \sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$.

The following series: $$\sum_{k=0}^{\infty}\frac{k}{2^{k+1}\left ( k+1 \right )^2}$$ came up as an intermediate step of calculating an integral. The answer according to Wolfram is $\displaystyle \...
3
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0answers
28 views

Convergence range of $\sum_{n=0}^{\infty}\frac {(-1)^n3n}{(x-4)^n}$

Find the range of convergence of $\displaystyle\sum_{n=0}^{\infty}\frac {(-1)^n3n}{(x-4)^n}$ Finding the radius first, setting $t=x-4$, $R=1/|((-1)^n3n)^{1/n}|=1$ So $\mid \frac 1 {x-4}\mid<1\...
0
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1answer
26 views

Summation simplification explanation

I'm trying to understand summation for my algorithm course and it has been a while since I took discrete math. Could any body please explain how does summation simplification work from the problem ...
1
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1answer
51 views

Can a series of polynomials converge non-uniformly?

Is there an example of a series of polynomials, say, the degree equals the index and converges non-uniformly? In other words, does point-wise convergence of a polynomial series imply uniform ...
5
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2answers
146 views

How to choose a contour in order to use the residue theorem to sum up a series from Ryzhik?

I would like to know how to sum up to following series (from the Gradshteyn-Ryzhik tables): $$\sum_{n=-\infty}^\infty\frac{e^{in\alpha}}{(n-\beta)^2+\gamma^2}=\frac{\pi}{\gamma}\frac{e^{i\beta(\alpha-...
3
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4answers
41 views

Convergence of the series: $\sum_{n\geq1}\frac{(-1)^n\arctan (n)}{n+n^{1/2}}$

$$\sum_{n=1}^{\infty} (-1)^n \tan^{-1}(n)/(n+(n)^{1/2}).$$ I know that the series is not absolutely converges. I want to prove using Alternative test. I don't know how to prove that sequence $ tan^{...
0
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1answer
71 views

A low-discrepancy or quasirandom series which would guarantee all value sequences

I am trying to find a type of quasi-random sequence which would guarantee that it could produce all possible sequences of values within the possible value range, while still producing random-seeming ...
4
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1answer
92 views

How would one go about integrating $x^x$?

So I have been trying to solve $\int x^x dx$ as a challenge from one of my friends. Before anyone says it, I recognize that the equation has no closed solutions. I have been trying to integrate by ...
2
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1answer
61 views

Evaluate $\lim_{n\to\infty} \frac{\sum_{r=0}^n\binom{2n}{2r}3^r}{\sum_{r=0}^{n-1}\binom{2n}{2r+1}3^r}$.

Evaluate : $$\lim_{n\to\infty} \frac{\sum_{r=0}^n\binom{2n}{2r}3^r}{\sum_{r=0}^{n-1}\binom{2n}{2r+1}3^r}$$ The answer given is $\sqrt3$. Frankly, have no clue where to begin. I thought of putting ...
0
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1answer
18 views

Unsure about expansion

Hello, can someone tell me how this expression is expanded in this proof. Does it follow from some other theorem?
0
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2answers
42 views

Help on Geometric Sequence Problem?

The sum of an infinite geometric series with first term a and common ratio r < 1 is given by $ S_n=a\cdot\dfrac{r^n-1}{r-1} $. The sum of a given infinite geometric series is $S_{\infty}=200 $ and ...
3
votes
2answers
116 views

If $g$ is continuous then $x^ng(x)$ converges on $[0,1)$

Suppose $g:[0,1]\to\mathbb R$ is a continuous function satisfying $g(1)=0$. Prove that the functions $f_n(x)=x^ng(x)$ converge uniformly on $[0,1]$. Hence or using Mean Value Theorem, prove that if $\...
0
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2answers
51 views

limit of sequence of quotients of sequence that converges

Let $$\lim_{n\to \infty}x_n=a$$ Prove that if $$\lim_{n\to \infty}{x_{n+1}\over x_n}=L$$ so $$|L|\le1$$ .... I tried for a long time but i can't prove that. please give me just a hint? thanks
2
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1answer
68 views

Bound for the sum of a finite sequence

Consider ${\bf c} = (a,b) \in \mathbb{R}^2$ with $0< \|{\bf c}\| < 1.$ Let $n \in \mathbb{N} $ and define \begin{align*} F_{n}(k) & := \frac{ [a + x_{n}(k)]^2}{ [a + x_{n}(k)]^2 + [b + y_{n}...
1
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1answer
39 views

Evaluate $A=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \cos(n x) dx$.

Evaluate the limits $$A=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \cos(n x) dx$$ and $$B=\lim_{n\to\infty} \int_a^b \operatorname{cotan}\left(\alpha x\right) \sin(n x) dx$$ ...
2
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2answers
531 views

When is it allowed to take a constant out of a series?

When is it allowed to take a constant out of a series? Suppose we have a series $\sum ca_n$, when can we write it as $c\sum a_n$? It's pretty obvious when we know beforehand the series converges ...
5
votes
1answer
214 views

Sum the infinite series

How to solve this: \begin{equation*} \sum_{n=1}^{\infty }\left[ \frac{1\cdot 3\cdot 5\cdots \left( 2n-1\right) }{ 2\cdot 4\cdot 6\cdots 2n}\right] ^{3} \end{equation*} I can make the bracket thing, $...
1
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2answers
55 views

Proving the series doesn't converge: $\sum_{n=1}^{\infty}a_n$, $\lim_{n\to\infty}na_n=\infty$, $a_1=-1$

Let $\displaystyle\sum_{n=1}^{\infty}a_n$ and $\displaystyle\lim_{n\to\infty}na_n=\infty$ and $a_1=-1$. Prove the series does not converge. From the given that $a_1=-1$ we know that there has to be ...
0
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1answer
70 views

Maximise $y$ with respect to $x$ for $y=\prod_{k=1}^{\infty}(1-x^{-k})$

$$y=\prod_{k=1}^{\infty}(1-x^{-k})$$ I want to maximise this function. So far I have: $$\ln(y)=\sum_{k=1}^{\infty}\ln(1-x^{-k})$$ $$\frac{dy}{dx}\cdot\frac{1}{y}=\sum_{k=1}^{\infty}\frac{kx^{-k-1}}{1-...
0
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2answers
35 views

divergence of a sum

Let f(x) = xsin$(\frac{\pi}{2x})$, x$\in(0,1]$. Consider the sequence $(x_i) = \frac{1}{2n - (i-1)}$, i = 1,...,2n. Show that $\sum_{i=1}^{2n}|f(x_i)-f(x_{i-1})|$ diverges as n goes to infinity. ...
0
votes
3answers
75 views

Convergence of sequence $x_{n+1}=\sqrt{2x_n+3}$

Let $f:\Bbb R \rightarrow \Bbb R$ be the function $f(x)=\sqrt{2x+3}$. (a) Show that for all $x\in[1,3], f(x)\geq x$. You may use the intermediate value theorem if you like. Let $x_0=1$ and ...
1
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1answer
24 views

$\lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0$

For $x \in ]0,1[$, how can one see that: $$\lim_{n \to \infty} 2^{n + 2} \times (x^{2^n} - x^{2^{n+1}}) = 0$$ Thanks.
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1answer
60 views

Infinite series to definite integral

Is there a method to transform the following infinite series to a definite integral? The problem is from my 1989 textbook on Calculus by Thomas and Finney, Ch 7., which focuses on methods of ...
4
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1answer
241 views

Help finding a closed form

I have the following function: $$\frac{2e^x}{e^{2x}+1+2x}=\sum_{n=0}^\infty \varepsilon_n\frac{x^n}{n!}$$ I would like to find a closed form for the $\varepsilon_k$. One thing that I do know is that ...
7
votes
4answers
194 views

What is the limit as k approaches infinity of $(k!)^{\frac{1}{k}}$ [duplicate]

What is the value of $$\lim_{k\to\infty}(k!)^{\frac{1}{k}}?$$ One of my students concluded the limit was infinity – which I tend to agree with, but was unable to show that was the limit. We ...
3
votes
2answers
123 views

Convergence of $\sum \frac{a_n}{1+a_n}$ when $\sum a_n$ and $\sum a_n^2$ converges.

Suppose $a_n$ are real numbers and $\sum a_n$ and $\sum a_n^2$ converges. How would one go about showing that $\sum \frac{a_n}{1+a_n}$ converges? ($a_n \neq -1$ for every $n$)
2
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4answers
62 views

Finding what $p$ does the series: $\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges

For what $p$ does the series: $\displaystyle\sum_{n=1}^{\infty}\frac{\ln n}{n^p}$ converges? My attempt: I wanted to use the limit comparison test and compare it with $\frac 1 {n^p}$ but it doesn't ...
1
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1answer
153 views

Number of apples in a basket riddle

You have six baskets with apples - 10,12,15,20,22,25 (this is how many apples there were in them - 10 in first, 12 in second..). Some of the apples are red and some are green. After one basket was ...
1
vote
1answer
212 views

What is the sum of this series involving factorial in denominator?

$$1 + \frac{1^2 + 2^2}{2!} + \frac{{1}^2 + {2}^2 + 3^2}{3!} + \cdots$$ I can't figure out how to do summations which involve a factorial term in the denominator. Please help. This is a past year ...
0
votes
1answer
25 views

Formula for combinatorial series sum [duplicate]

As a part of one computer algorithm, I want to find sum for $$n+ \frac{n(n+1)}{2!} + \frac{n(n+1)(n+2)}{3!}+....+ \frac{n(n+1)(n+2)...(n+r-1)}{r!} $$. I looked at $$\frac1{(1-x)^n}$$. But it is ...
1
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2answers
130 views

Convergence of averging series

If $\sum\limits_{n=1}^\infty a_n$ converges where $a_n>0,\ \forall n\in\mathcal N$, prove $\sum\limits_{n=1}^\infty \sqrt[n]{a_1a_2\cdots a_n}$ and $\sum\limits_{n=1}^\infty\frac{n}{\frac{1}{a_1}+\...
-4
votes
2answers
42 views

what's the limit of the following sequence? [closed]

$\displaystyle \frac{n^n}{3^nn!}$ Already tried to simplify the fraction but i can't solve the indetermination, please show me how to solve it step by step
2
votes
1answer
30 views

Problem related to means

I am confused on proving that If a, b, c are 3 numbers in harmonic progression Then ${(a^n+c^n) /2} >({(a+c)/2}) ^n$ I attempted like this... Since a, b, c are in hp so $(a+c) /2>b$ ...
1
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0answers
50 views

Strange sequence needed

There's no easy way to explain this, but please bear with me. I'll try to keep it slow and simple. I'm looking for a property that is related to the generalised pentagonal numbers (A001318 in the ...
0
votes
1answer
74 views

Where does this equation come from: $ (1+mx)^n = 1 + \sum_{n=1}^{\infty} {\binom{2n}{n} \over 4^n } x^n $

I have found the following problem here: https://brilliant.org/problems/intriguing-sum/?group=Km7yEIDGtHDa&ref_id=709399 In the solution a solver directly started with the equation given in the ...