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

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0
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0answers
18 views

Proving that a sequence fits a certain formation rule.

I've got this function here: $$F(m):=-\frac{\sum _{i=1}^{m-1} \text{F}(i) \left(\binom{m}{i-1}-3^{m-i}\right)}{m-1}$$ $$F(1):=x$$ and if I calculate the values for $m = {1,2,3,4,\ldots,n}$ I get ...
0
votes
1answer
57 views

Does the series $\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$ converge?

Does the series $$\sum_{n=0}^\infty\frac{\sin(n+\frac 12)\pi}{1+\sqrt{n}}$$ This is supposed to be an alternating series but I can't seem to figure out what the $b_n$ is in this case. is there some ...
1
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4answers
43 views

Convergence of a series involving arcsin

I need to determine the convergence of the following series. I tried a few tests but they turned inconclusive. $\sum_{n=1}^\infty\arcsin \ 1/2^n$
0
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4answers
59 views

Find the sum of the following series $\sum_{k=0}^\infty \frac{4^k-3^k}{5^k}$

I need to find the sum but am struggling to figure out the correct approach. $$ \sum_{k=0}^\infty \frac{4^k-3^k}{5^k} $$
0
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1answer
36 views

What would describe the following basic sequence?

I found the following sequence in a programming tutorial and thought it look neat and noticed the pattern, but am mathematically challenged and can not quite describe it algebraically. Here is what it ...
4
votes
1answer
60 views

if $a_{n+1}=\sqrt{a^2_{n}-2a_{n}+2}-1$, show that there exists a constant $c$ such $a_{2n}<c<a_{2n+1}?$

let sequence $\{a_{n}\}$,such $a_{1}=1$,and $$a_{n+1}=\sqrt{a^2_{n}-2a_{n}+2}-1$$ prove or disprove :there exists a constant $c$ such $$a_{2n}<c<a_{2n+1}?$$ my idea: since ...
3
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0answers
68 views

Closed-form of $\sum\limits_{n=1}^{\infty}\dfrac{x^n}{\sqrt{n\pi}}$

What is the closed-form of the following series $$\sum\limits_{n=1}^{\infty}\frac{x^n}{\sqrt{n\pi}}\ ?$$ From a different approach I got the answer as $(1-x)^{-1/2}$, by using the fact that ...
2
votes
3answers
52 views

prove ${a_n}$ is a Cauchy sequence, provided $a_{n+2} = \frac{a_n + a_{n+1}}{2}$ [closed]

Suppose there is a sequence with the property $$a_{n+2} = \frac{a_n + a_{n+1}}{2}$$ for all $n \in \mathbb{N}_{+}$ Prove that ${a_n}$ is a Cauchy sequence I've self-taught ...
0
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2answers
84 views

Reduce the expression $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}$ into a Geometric series [closed]

Is there a way to reduce the expression $$\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}}$$ into a Geometric series.
1
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0answers
72 views

Closed form of $\sum_{n=1}^{\infty} n^{-n}$

What is the sum of the following series? $$\sum_{n=1}^{\infty} n^{-n}$$ It definitely converges by ratio test.
8
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0answers
61 views

A sequence that avoids both arithmetic and geometric progressions

Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link. I started to explore sequences ...
4
votes
2answers
71 views

Show that the following series is less than $4 \pi^2 / 3$.

Show: For any $k = 0,1,2,...$, $$ \sum_{i=0}^{i=k} \frac{(k+1)^2}{(i+1)^2 (k-i+1)^2} \leq \frac{4 \pi^2}{3}. $$
6
votes
2answers
507 views

Sum of this series

$$ \mbox{How do I find the sum of this series}\quad \sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}\ {\large ?} $$ Hints in the right direction would be appreciated.
0
votes
0answers
16 views

Limit of a sum over an increasing finite set: dominated convergence / riemann integral

Let $(S_k)_{k\in\mathbb{N}}$ be a sequence of finite sets where $S_k \subset [0,1]$ for all $k$. It is assumed that $S_k\subset S_{k+1}$ and that $$\lim_{k\to\infty }\max_{s\in S_k} (s-s')=0$$ where, ...
2
votes
1answer
39 views

Alternating test

I ran into some alternating test somewhere, and I had difficulty applying it directly (It's not an HW and I'm not a student) : $$ \sum_{ n \geq 1 } (-1)^n \frac{n^3+n+1}{\sqrt{2n^7+n^5+1}} $$ The ...
0
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0answers
36 views

Sum of absolute value of cos(an+b)? [closed]

Can anybody help me on deriving this? where, a and b are constants. This converges since N is finite, and the converged values for most of a and b are close each other for sufficiently large number N. ...
14
votes
5answers
155 views

For $a_n,b_n\uparrow$ and $\sum \frac{1}{a_n}$, $\sum \frac{1}{b_n}$ divergent is the series $\sum \frac{1}{a_n+b_n}$ also divergent? [duplicate]

Let $a_n$ and $b_n$ are strictly increasing to $+\infty$ sequences such that the series $\sum \frac{1}{a_n}$ and $\sum \frac{1}{b_n}$ are divergent. Is it true that the series $\sum \frac{1}{a_n+b_n}$ ...
1
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4answers
45 views

Express $3.72444\ldots$ as a fraction using the formula for geometric progressions

I did the following: $$3.72\overline{4} = 3.724+\left(\left(\frac{4}{10^4}\right)+\left(\frac{4}{10^5}\right)+\left(\frac{4}{10^6}\right)+\cdots\right)$$ where $a=3.724$ and $r=\dfrac{1}{10}$ Using ...
4
votes
1answer
64 views

Yet another limit

Let's $x_n\ge0$ and $$\overline{lim}_{n\rightarrow\infty}\frac{x_1+x_2+\cdots+x_n}{n}\lt+\infty,~~\lim_{n\rightarrow\infty}\frac{x_n}{n}=0.$$Prove that ...
3
votes
1answer
36 views

Total number of possible sub sequence with given condition

Given a sequence of two letters A and B find the total number of possible sub sequences where number of letter A is two times the number of letter B without ...
2
votes
3answers
164 views

finding explicit formula

The question ask us to guess an explicit formula for the sequence $$y_k = y_{k-1} + k^2 ,$$ for all integers $k$ greater than or equal to 2 and $y_1 = 1$ Can someone help me with this? so far what ...
1
vote
1answer
12 views

finding minimum common element in two arithmetic sequences

i have two different arithmetic sequences and i know their first term and common difference. Is there any short technique or some formula by which i can find the first element which is common in both ...
1
vote
3answers
61 views

Can you show that the LHS equals the RHS in this equation, by showing how I can get the expression on the RHS?

$$ \frac{1^2+2^2+...+(n-1)^2}{n^3} = \frac{(n-1)n(2n-1)}{6n^3} $$ Can someone show me step by step how I can transform the LHS to the RHS? If possible, using high school-level math. I have now ...
1
vote
4answers
50 views

finding explicit formula through substitution method

The question ask us to guess an explicit formula for the sequence $$t_k = t_{k-1} + 3k + 1 ,$$ for all integers $k$ greater than or equal to 1 and $t_0 = 0$ Can someone help me with this? Because I ...
3
votes
4answers
151 views

What is the sum of this series: $\displaystyle\sum_{n=0}^{\infty} \dfrac{a^n}{(3n)!}$ [duplicate]

I tried getting it into a closed form but failed. Could someone help me out? $$\sum_{n=0}^{\infty} \dfrac{a^n}{(3n)!}$$
1
vote
1answer
52 views

simplifying the series $\sum\limits_{n=0}^{s-1}a^n/n! $

$$ \sum_{n=0}^{s-1}a^n/n! $$ Is there a nice way to simplify this series assuming $0 < a <1$ ?
1
vote
2answers
30 views

Guess explicit formula using iteration

The question ask us to guess an explicit formula for the sequence $$s_k = s_{k-1} + 2k ,$$ for all integers $k$ greater than or equal to one and $s_0 = 3$ Can someone help me with this? Because I ...
1
vote
1answer
67 views

Convergence of a series involving cotangent function

Is the following series convergent or not and why? $\sum_{n = 1}^{\infty} \cot(\pi/2 - 1/n)$. I don't know why I cannot get this, but I was expecting either $\cot(\pi/2 - 1/n) < x^{2}$ or ...
3
votes
1answer
30 views

Property of Subadditive Sequence

Call a sequence $\left \{ a_n \right \}$, $n \geq 1$, strictly subadditive if it satisfies the inequality $$ a_{n+m} < a_n+a_m $$ for all $m$ and $n$. I am wondering whether it is necessarily true ...
0
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3answers
59 views

On a certain series of complex numbers

Is it possible that the above infinite series is equal to ?
3
votes
1answer
83 views

Infinite product: $\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}$ [duplicate]

I am trying to find $$\lim_{n\rightarrow \infty}\prod_{k=2}^{n}\frac{k^3-1}{k^3+1}.$$I have no idea how I can start. Please help me. Thanks!
-1
votes
3answers
74 views

Can there be more than one power series expansion for a function.

I guess the answer is NO, for polynomials. I know that there are more than one series expansion for every function. But I am talking about power series here. All Ideas are appreciated
0
votes
1answer
40 views

$\sum_{n=0}^{\infty} n \left( \frac{2}{3} \right)^n = ?$ [duplicate]

$$\sum_{n=0}^{\infty} n \left( \frac{2}{3} \right)^n = ?$$ How to find it? If it lacked n before fraction, I would use formula for the sum of geometric series.
0
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2answers
36 views

It is a question about uniform convergence of a function.

I solved this problem . Is my answer a correct ? $$ x \in [0,\infty) ,\lim_{n \to \infty} \frac{nx}{1+n^2x^2}=0\ \ \ $$ Is $ \frac{nx}{1+n^2x^2} $ converged uniformly on $0$ ? My solution $$ ...
2
votes
0answers
76 views

proof question about an infinite sums [duplicate]

I once saw a place one interesting question, the question is: proof that if $\displaystyle\sum_{n=1}^{+\infty}a_n$ diverges, then $\displaystyle\sum_{n=1}^{+\infty}\frac{a_n}{a_1+\cdots+a_n}$ ...
8
votes
3answers
149 views

The limit of a sequence $\lim_{n\rightarrow \infty}\prod_{k=0}^{n-1}( 2+\cos \frac{k\pi }{n})^{\pi/n}$.

$$ \lim_{n \to \infty}\prod_{k = 0}^{n - 1} \left[\,2 + \cos\left(k\pi \over n\right)\right]^{\pi/n}\ =\ ? $$ Please give some hints.
0
votes
2answers
68 views

Minimum of $f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$

Show that $$f(x,y)=\sum_{n=0}^{+\infty}\frac{(n^2−nx−y)^2}{2^n}$$ is defined on $\Bbb{R}^2$, it has a minimum and find for which couple $(x, y)$ the minimum is reached. The first point is okay, ...
0
votes
0answers
20 views

Rearrangements of Dirichlet Eta Function

I was wondering if explicit closed forms for rearrangements of $\eta(s)$, for $s$ such that the series is not absolutely convergent, are useful in studying the Dirichlet $\eta$ function. I am asking ...
9
votes
2answers
105 views

Divergent of a vector field on a sequence of spheres

I'm studying for my exams and I found this problem in the book "Advanced Calculus", written by Friedman: "Consider a sequence of spheres $S_n$ in $\mathbb{R}^3$ with center $P_n$ and radius $r_n$, ...
2
votes
2answers
26 views

Algebraic Symbol Manipulation While Finding the Sum of a Series

In a precalc text, in the chapter on geometric progressions and series, we're told of the formula $S=\frac{a(1-r^{n+1})}{1-r}$ and that: $S=\frac{a}{1-r}$ is valid for $|r|<1$ We're then asked ...
3
votes
1answer
60 views

Limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ - help finishing proof…

EDIT: Fixed geometric proof off-by-one error. I am looking for the limit of $x_n = 0.5(x_{n-1} + x_{n-2})$ with $x_2 > x_1$ arbitrary. I can show $\forall n: |x_{n}-x_{n+1}| = \frac{c}{2^{n-1}}$ ...
1
vote
1answer
58 views

Convergence of $\sum_{n=1}^{\infty} n! (4x - 5y)^n $

Where does the following series converge? $$\sum_{n=1}^{\infty} n! (4x - 5y)^n $$ Any (x,y) from the line $4x=5y$ gives series that consists of zero terms, hence, converges. I need help in ...
1
vote
0answers
48 views

Fourier series using summation methods

My question is similar to this one. There are ways of deriving the formulae like $$\sum_{k = 1}^\infty \frac{\sin(kz)}{k} = \frac{\pi - z}{2}$$ using summation methods. My question is: How can we ...
1
vote
1answer
78 views

Find limit $\lim\limits_{x \to \infty} \int_0^{x} \cos\left(\dfrac{\pi t^2}{2}\right)$

I looked at the graph and found that limit is $\dfrac{1}{2}$ And limit to $-\infty$ is $-\dfrac{1}{2}$ By the way, the function for which we are finding the limit is called Fresnel function
0
votes
1answer
38 views

a finite induction question from burton's elementary number theory

this question comes from burton's elementary number theory, 4th edition. question 3 in 1.1 says to use the second principle of finite induction to establish that $$for\ all\ n\ge1,\ ^{(a)}\ ...
0
votes
2answers
31 views

Filter on Fourier Series

i have a lowpass filter H(ω) which is $ H(ω) = e^{-jω} $ on -2π≤ω≤2π, and $0$ elsewhere and i have a function in fourier series y(t), i need to find the new signal (z(t)) after the application of the ...
3
votes
1answer
63 views

Limit of a recursiv trigonometric function

So while doing my math homework I recently stumbled upon this little thing: It seems that $y_n = \sin(y_{n-1}) + k$ with arbitrary $y_0$ and $k$ converges to a definite value for $n \to \infty $. ...
0
votes
0answers
43 views

How can I cleverly use the reverse triangle inequality in this case?

Say, we have the following recurrence relation: $$x_{k+2}=4x_{k+1}+x_k(-3-2h\lambda)$$ with $x_0 $ given, $x_1=(1+h\lambda)x_0, h$ small (step size), and $\lambda<0$. Here is the context is case ...
4
votes
2answers
111 views

Closed form of $\displaystyle\sum_{n=1}^\infty x^n\ln(n)$

Is there a closed form of this : $$\sum_{n=1}^\infty x^n\ln(n),$$ where $|x|<1$. Thanks in advance.
5
votes
2answers
127 views

Evaluating a sum involving binomial coefficient in denominator

I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$ I thought that this can be evaluated using the expansion of ...