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

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2
votes
3answers
109 views

If $\sum x^6_n$ converges, then $\sum x^7_n$ converges too

If $\displaystyle\sum x^6_n$ converges, then $\displaystyle\sum x^7_n$ converges too.
2
votes
1answer
35 views

Need to know why $\sum_{k=0}^{\infty}kr^{k} = \frac{r}{(1-r)^{2}}$

Working on a Stat problem where I must find $E(x)$ of $f(x)=\left(\frac{1}{2}\right)^{x+1}$ for $x=0,1,2,\cdots$ I have, ...
2
votes
2answers
35 views

Harmonic Series question about convergence

For large enough $n \in \mathbb{N}$, consider the sequence $(a_i (n))_{i \in \mathbb{N}} \overset{\Delta}{=} (a_i)_i : a_i(n) \overset{\Delta}{=} a_i = \frac{\sum_{x=1}^n (\frac{1}{x^i})}{i} \forall i ...
2
votes
1answer
39 views

Convergence of $\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$

I need help determining what following series converges to using the ratio test. $$\sum_{n=0}^{\infty} \frac{3^{2n}}{(2n)!}$$ It's the end that really has me confused with what to do with the ...
1
vote
2answers
44 views

Question on the sum $\sum_{n=1}^{\infty}\frac{x^n}{n} = -\ln(1-x)$

$f(x) = \displaystyle\sum_{n=1}^{\infty}\frac{x^n}{n} = x + \frac{x^2}{2} + \frac{x^3}{3} + ... = -\ln(1-x)$ for $|x| < 1$. $f'(x) = \displaystyle\sum_{n=1}^{\infty}x^{n-1} = 1 + x + x^2 + x^3 ...
0
votes
1answer
34 views

A sequence converges if and only if every subsequence converges?

I want to prove this and intuitively it makes sense. But I'm having a hard time coming up with a proof. So if a sequence converges, then we have a natural number for which the distance between all ...
1
vote
1answer
38 views

Calculate $\sum_{n=1}^\infty\int_{\sqrt{3+n^{1/5}}}^{\sqrt{4+n^{1/5}}}\frac{e^{t^2}}{(e^{t^2}+1)t^2}dt$.

Check the convergence of the series $$\sum_{n=1}^\infty\int_{\sqrt{3+n^{1/5}}}^{\sqrt{4+n^{1/5}}}\frac{e^{t^2}}{(e^{t^2}+1)t^2}dt$$ Any suggestions please? I can't calculate integral! Thanks in ...
-1
votes
0answers
45 views

Calculus - comparison test for series [closed]

Prove that if $b_n\geq 0$ and $$\lim_{n\to\infty}\frac{a_n}{b_n} =q\in(0,+\infty)$$ then the series $\sum a_n$ is convergent iff the series $\sum b_n$ is convergent. Use this fact to determine the ...
1
vote
1answer
26 views

Non-vanishing terms of Maclaurin series for $\log(3-\cos x^2)$

I have to find the first two non-vanishing terms in the Maclaurin series of $$g(x) = \log(3 − \cos(x^2))$$ and that prove $x=0$ is a stationary point. What is a quick way of working out the ...
4
votes
2answers
47 views

Integral with series

How do I represent this integral $$\int_{0}^{1} \frac{10}{10+x^4} dx$$ as a series so that I can calculate with an error of less than $10^{-5}$.
1
vote
1answer
20 views

The Convergence of a Complex Valued Infinite Series

Check the convergence and calculate the radius of convergence of the series $$ \sum^{\infty}_{n=1}\frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}z^n,\forall\alpha\in\mathbb{C}. $$ I tried to use the ...
1
vote
1answer
36 views

Converging series

Suppose that $\sum_{n=0}^\infty a_{n}$ is a convergent series, with $a_{n}\gt0$ and suppose that $b_{n}\gt0$ is a bounded sequence. Then show that the series $\sum_{n=0}^\infty (a_{n}b_{n}$) is ...
2
votes
1answer
22 views

Find the first two non vanishing maclaurin terms

Find the first two nonvanishing terms in the Maclaurin series of $\sin(x + x^3)$. Suggestion: use the Maclaurin series of $\sin(y)$ and write $y = x + x^3$ Using this result, find ...
3
votes
1answer
45 views

show that the series converges.

Using the definition, show that the series $\sum_{n=1}^\infty \frac{1}{n(n+3)}$ converges. Determine the limit. This is what I've managed to do so far: $\frac{1}{n(n+3)}$ = $\frac{1}{3n}$ ...
1
vote
3answers
31 views

Computing the trigonometric sum $ \sum_{j=1}^{n} \cos(j) $

I have a task to compute such a sum: $$ \sum_{j=1}^{n} \cos(j) $$ Of course I know that the answer is $$ \frac{1}{2} (\cos(n)+\cot(\frac{1}{2}) \sin(n)-1) = \frac{\cos(n)}{2}+\frac{1}{2} ...
1
vote
2answers
32 views

Find the value of $2p+4q+7r$ given that $2p,\ q,\ 2r$ are in geometric progression.

It is given that $2p,\ q, \ 2r$ are in G.P. Also the roots of the quadratic equation $$px^2+qx+r=0$$ are of the form $\alpha ^2,\ 4\alpha -4$. Find the value of $2p+4q+7r$. From the given data: ...
3
votes
1answer
55 views

Limes of $a_n = i^n$

Out of couriosity and for my understanding i want to ask: When i have the sequence $a_n = i^n$ While i is the imaginary number, i will of course have four accumulation points: $-1,1,-i,i$. So the ...
-1
votes
3answers
31 views

Sequence limit… [closed]

I need your help to find the limit of this sequence $$\sum_{k=1}^{n}\frac{1}{n+k}$$ We know that: $$\ln\frac{x+1}{x}\leq \frac{1}{x} \leq \ln\frac{x}{x-1}$$ Thank you in advance.
3
votes
2answers
172 views

Concerning alternating series: test for divergence fails (typo in the book).

Here is a series: $$ - \frac 2 5 + \frac4 6 - \frac 6 7 + \frac8 8 - \frac{10} 9 +\dots $$ The series is convergent (it says so in the back of the book) but the test for divergence fails: We have, ...
0
votes
2answers
33 views

Converging series of functions

I don't know even how to think about it: Let $f$ be defined in the interval $[-1, 1]$. Assume $f(0)=0$, its derivative is continuous in $(-1,1)$ and $f'(0)=\beta \neq 0$. Prove that ...
0
votes
0answers
28 views

Solutions of fractional linear dynamical systems

The Mittag-Leffler function is defined as: $$ E_\alpha(\tau) = \sum_{k=0}^{\infty}\frac{\tau^k}{\Gamma(\alpha k + 1)}, $$ which can also be defined, analogously, for matrices $A\in\mathbb{R}^{n\times ...
0
votes
1answer
59 views

In ΔABC prove that sides are in $AP$. [closed]

In ΔABC, if $$\tan\left(\frac{A}{2}\right)=\frac{5}{6}$$ and $$\tan\left(\frac{B}{2}\right)=\frac{20}{37}$$ prove that sides lengths $a$, $b$ and $c$, when sorted, form an arithmetic progression My ...
1
vote
3answers
56 views

Prove if $\left\{x_n\right\}$ converges to $2$, then $\left\{\frac{1}{x_n}\right\}$ converges to $\frac{1}{2}$

We know that for all $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that $\lvert x_n - 2 \rvert < \epsilon$ for all $n \geq N$, and we want to show that for all $\varepsilon' > ...
4
votes
2answers
43 views

Cesáro sums and the actual limit

My textbook, as an aside, defines the Cesáro sum as follows: $$ \sigma_n= \frac{s_1+...+s_n}{n}= \frac{1}{n}\sum_{k=1}^ns_k, $$ where $$ s_n = \sum_{k=1}^na_k. $$ This method is used, I am told, to ...
0
votes
1answer
37 views

How find this Exponential constant $C$,if such this $Ax^C\le N(x)\le Bx^C$

interesting problem Let sequence $$a_{0}=x\in (0,1),a_{n}=a_{n-1}+a^3_{n-1},n=1,2,\cdots$$ and define $$N(x)=\min{\{n|a_{n}>1\}}$$ Assmue that there exsit postive constant $A,B$,and ...
0
votes
2answers
40 views

Convergent Series (Comparison Test)

I'd like to show $\sum_n a_n$ converges if and only if $\sum_n \frac{a_n}{1+a_n}$ converges. Where each $a_n$ is a sequence of positive real numbers. The first side is trivial since $a_n > a_n / ...
0
votes
1answer
32 views

Show that $(α^{1/u_{n+1}}-1)^{1/(n+1)}<(α^{1/u_{n}}-1)^{1/n}$

Let $α>2$ be a real number. Let $(u_{n})_{n}$ be an increasing sequence. Then my question is: Show that $$(α^{1/u_{n+1}}-1)^{1/(n+1)}<(α^{1/u_{n}}-1)^{1/n}$$ Add. We have: ...
0
votes
0answers
24 views

Series diverging to infinity and series converging to infinity - is there a difference?

There is a lot of diverging series which can be this or that way summed up to get a finite value. There are also series whish infinitely grow, but nevertheless can be summed up by a certain technique. ...
0
votes
3answers
25 views

Proof for result of sum of 3 elements of recursive sequence

I have a recursive sequence: $$a_1=1\\a_2=1\\a_3=-1\\a_k=a_{k-1}\cdot a_{k-3} (for\,k>3)$$ So this sequence has cycle of 7: $1,1,-1,-1,-1,1,-1$ And I have to calc $a_{2013} + a_{2014} - ...
6
votes
1answer
604 views

Given two real sequences that go to infinity, is it possible to select two subsequences that grow at the same rate asympotically?

Given two positive real sequences $a_n$ and $b_n$ that both diverge to infinity, is it possible to choose two subsequences $a_{s_n}$ and $b_{t_n}$ such that $a_{s_n}/b_{t_n}\rightarrow1$?
3
votes
1answer
35 views

Series representation

I have started this problem by expanding it so that i can get some cancellation term, but couldn't reach on the correct result.I got the result like -ln4 -ln5-ln6..... Please have a look on this.
1
vote
2answers
31 views

Non-harmonic-like series that diverges even though $a_\infty=0$

The only series that I can think of is the harmonic series or the harmonic series with primes, which are well known. Are there any other series of this type?
1
vote
1answer
31 views

Weierstrass M-test help

I am supposed to use M-test on this one $$\sum \frac {n\ln (1+nx)}{x^n}$$ on $$1<x< \infty$$ But I face problems finding an appropriate $M_n$, thanks for help
1
vote
1answer
20 views

Prove that $f_{n}:[1,+\infty[ \rightarrow \mathbb{R} : x \rightarrow \frac{x^n}{1+x^{n}}e^{-x}$ is increasing

By definition, I want to prove that $f_{n+1}(x)-f_{n}(x)\ge0$ for $x \in [1,+\infty[$ So, we obtain : $e^{-x}(\frac{x^{n+1}-x^{n}}{(1+x^{n+1})(1+x^{n})})$ But for $x \in [1,+\infty[$ we have : ...
10
votes
3answers
48 views

Is this sequence a recurrence relation?

$21, 36, 55, 60, 67, 68, 92, 93, 125$ I thought maybe it's $T_{2n + 4} + 2(n + 4)$ and I tried a few other formulas involving triangular numbers. I also tried variations on the Fibonacci sequence, ...
1
vote
1answer
43 views

which convergence test is better?

I need to check this one for absolute convergence $$\sum^{\infty}_1 \frac {(-1)^n(n+4)}{(n^2+1)^{1/4}(2+\sqrt{n^2+3})}$$ But I am not sure which method to use, it fails with Root or Ratio tests.
0
votes
1answer
25 views

Does the following series converges uniformly?

$\sum \frac{x^2}{n^2}$ where $x \in [5,\infty)$ clearly we can't apply the Weierstrass M test because as $x \longrightarrow \infty$ we can't say that $\sum \frac{x^2}{n^2} \leq \sum \frac{1}{n^2}$ ...
2
votes
4answers
52 views

Why is $\frac {1\cdot2\cdot3\cdot…\cdot n}{(n+1)(n+2)…(2n)}\le \frac 1 {n+1}$

I'm trying to find the limit: $\displaystyle\lim_{n\to\infty}\frac {1\cdot2\cdot3\cdot...\cdot n}{(n+1)(n+2)...(2n)}$ I thought of taking a pretty obvious binding from above expression: $\frac {n^n} ...
3
votes
0answers
40 views

Taylor series of $f(x) = \arctan(x)$ converges to $\arctan(x)$

I have to find out the Taylor series of $f(x) = \arctan(x)$ and prove that it converges to $f(x)$ for any $x \in (-1, 1) $. So far I determined the Taylor series to $T_f(x) = \sum ...
0
votes
3answers
44 views

Simply normal sequence of $-1$'s and $1$'s as coefficients of harmonic series

Suppose $s_{n}$ is either $1$ or $-1$ for $n=1,2, 3,\ldots$ and that half the $s_{n}$'s are 1; i.e. $$ \lim_{n\to \infty} \frac{\#\{i\leq n: s_{i}=1\}}{n}=\frac{1}{2}. $$ Does then the series ...
0
votes
1answer
17 views

Series convergence problem for positive x

Having the following sequence: $\sum\limits_{n=1}^\infty \frac{n!}{\left(x+1\right)\left(x+2\right) \ldots \left(x+n\right)}$ $x>0$ how to investigate its convergence? Which criteria should be ...
3
votes
1answer
27 views

proving a limit of a series with a sum [duplicate]

I just can't find a way to prove it.
3
votes
1answer
62 views

Is my proof that $\sum \frac{(-1)^n}{n+\frac{10100}{n}}$ is convergent valid?

I want to answer whether the following series is convergent or divergent: $$\sum_{n=1}^{\infty} \frac{(-1)^n}{n+\frac{10100}{n}}$$ Alternating series test seems like a good idea. So if I prove that ...
1
vote
3answers
31 views

How did we get $ p\sum_{n=1}^{∞} (1-p)^{n-1}=\frac{p}{1-(1-p)}$

I am not sure how we got below expression.. $$\sum_{n=1}^{∞} P(X=n)= p\sum_{n=1}^{∞} (1-p)^{n-1} = \frac{p}{1-(1-p)} = 1$$ I understand that we calculate expected value for n trials using linearity ...
2
votes
2answers
48 views

Find the minimum possible value of $x(1-z)+y(1-x)+z(1-y)$

It is given that $$xyz=(1-x)(1-y)(1-z)$$ and $$x, y, z \epsilon (0,1)$$ Find the minimum possible value of the expression: $$x(1-z)+y(1-x)+z(1-y)$$ Using the AM-GM inequality concepts, I can write ...
-1
votes
1answer
21 views

Show that partial sums of this series are bounded

Following series: $\sum\limits_{n=1}^\infty \sin n\phi=\frac{\sin\frac{n+1}{2}x\cdot\sin\frac{n}{2}x}{\sin\frac{x}{2}}$ now i need to show that partial sums of this series are bounded. How to do that? ...
0
votes
2answers
38 views

Does $\sum_{n=1}^\infty (-1)^n\frac{n\log n}{e^n}$ converge absolutely?

How to check whether the series converges absolutely: $$\sum_{n=1}^\infty (-1)^n\frac{n\log n}{e^n}$$ I tried tests like Ratio, Raabe but not working
0
votes
1answer
32 views

What is $\lim_{n\rightarrow \infty}\frac{1}{n}\sum _{k=1}^{\lfloor\frac{n}{2}\rfloor}\cos\frac{k\pi}{n}$

How to evaluate this: $$\lim_{n\rightarrow \infty}\frac{1}{n}\sum _{k=1}^{\lfloor\frac{n}{2}\rfloor} \cos\frac{k\pi}{n}$$ where $\lfloor\frac{n}{2}\rfloor$ denotes the largest integer not exceeding ...
1
vote
1answer
19 views

Convergence of a series with trigonometric functions

I have two sequences as follows: $\sum\limits_{n=1}^\infty \frac{\sin n\phi}{n}$ $\sum\limits_{n=1}^\infty \frac{\cos n\phi}{n}$ How to investigate convergence of those two? What criteria should i ...
1
vote
2answers
61 views

limit of a sequence with $\pi $ and $ e $ [duplicate]

Let $$ a_n = \frac{n}{\pi}sin(2\pi e n!) $$ Find $$\lim_{n\to\infty}a_n $$ I tried expanding using the Taylor series representation of $e$ but I arrived nowhere.