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

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3
votes
0answers
35 views

Are these Infinite Series Representations of Special Functions?

I am not sure how to google the answer for this question. Anyway, in trying to compute the velocity of a charged particle in an electromagnetic field, I came across these two infinite series ...
2
votes
4answers
48 views

$\sum (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$ convergent but not absolutely convergent

I need to prove that: $$\sum_{n=1}^{\infty} (-1)^{n+1}\log\left(1+\frac{1}{n}\right)$$ is convergent, but not absolutely convergent. I tried the ratio test: $$\frac{a_{n+1}}{a_n} = ...
1
vote
2answers
74 views

If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$

I have the following statement: If $\sum_{n=0}^{\infty}a_{n}x^n $ converges for $|x| < R$ , then $\sum_{n=0}^{\infty}na_{n}x^n $ converges for $|x| < R$ as well. I couldn't find a ...
1
vote
3answers
46 views

Expand a function to power series

I have the following function and i try to expand it to a power series - $$F(x) = \frac{2x}{(x^2+1)^2}$$ around $X = 0$ I tried to substitute $t = -x^2$ and got stuck. I would like to get some help ...
-2
votes
3answers
64 views

How is this series rearranged?

I'm stuck at this. How is RHS rearranged? Is it a change of index? $$ \sum_{n=1}^{2N} \frac{1}{n} - \sum_{n=1}^{N} \frac{1}{n} = \sum_{n=N+1}^{2N} \frac{1}{n} $$ I'm stuck here: $$ \sum_{n=1}^{2N} ...
0
votes
1answer
31 views

The sum of series in an interval

I have the following series - $$ \sum_{n=1}^\infty nx^{2n-1} $$ I found that its convergence interval is $[-1,1]$ but how can i calculate the sum in this interval ? i would like to get some hint ...
47
votes
2answers
2k views

What is the average rational number?

Let $Q=\mathbb Q \cap(0,1)= \{r_1,r_2,\ldots\}$ be the rational numbers in $(0,1)$ listed out so we can count them. Define $x_n=\frac{1}{n}\sum_{k=1}^nr_n$ to be the average of the first $n$ rational ...
0
votes
0answers
21 views

Prove the function $G_2^{(0,a)}$ is holomorphic

I fix $a\in \{1,2,3\}$. I am working on a proof and i need to prove that the function $G_2^{(0,a)}$: $$G_2^{(0,a)}(z):=\frac {\pi^2}8-\frac{\pi^2}2\sum_{k\ge 1}\ \sum_{j\ge 1}je^{i\frac \pi 2 ...
1
vote
0answers
53 views

Proving this formula for the Zeta function?

Could some one link me to a proof of this integral? $\zeta{(s)} = \frac{1}{\Gamma{(s)}}\int_{0}^{\infty} \frac{x^{s-1}}{e^x - 1} dx$ All the sites I've seen so far just introduce with the definition ...
0
votes
1answer
24 views

in which subset of $R^2$ the series is convergent?

For $(x,y) \in \Bbb R^2 $ ,consider the series $\lim_{n \to \infty } \sum_{l,k=o}^n \frac{k^2x^ky^l}{l !} $ .Then the series converges for $ (x,y)$ in 1.$(-1,1)\times (0, \infty )$ ...
3
votes
1answer
64 views

Determining convergence of a series $\sum_n (-1)^n \sin a_n $

I need to determine if the following series is convergent: $$\sum_{n=2}^\infty (-1)^n\sin\left(\frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n\ln^2(n)}\right).$$ I've tried to use alternating series test but ...
0
votes
0answers
37 views

Calculating sum of a ceil function involving golden ratio

Let $\tau$ be a function on natural numbers defined as $\tau(n)=\lceil n\phi^2\rceil$, where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio. Is there a way to calculate ...
1
vote
0answers
34 views

Looking for a closed form sum of a series

I am wondering if there is a closed form solution of the following series $$ 2 \sum_{n =1}^{\infty} \frac{\sin (\pi a n) }{\pi n} (1-\cos (\pi n)) \exp \left(-(\pi a n)^2 t^{2 b}\right) $$ where ...
1
vote
3answers
77 views

If $a, b, c >0$ prove that $ [(1+a)(1+b)(1+c)]^7 > 7^7a^4b^4c^4 $.

I solved it using AM, GM inequalities and reached to $[(1+a)(1+b)(1+c)]^7 > 2^{21}(abc)^\frac72 $ please help how to get $7^7(abc)^4$ in the inequality.
1
vote
0answers
21 views

About the uniform convergence of a series

Let $(a_n(z))$ is a sequence of holomorphic functions defined on $\mathbb{C}\setminus A$, where $A$ is a set of simple poles. I am thinking about proving that $\sum_{n=1}^{\infty}\left | a_n(z) ...
0
votes
2answers
57 views

$\sum_{n=1}^{\infty}\frac{1\cdot 4\cdot 7\cdots (3n+1)}{n^5}$

$$\sum_{n=1}^{\infty}\frac{1\cdot 4\cdot 7\cdots (3n+1)}{n^5}$$ this question comes right after the question that asks me to prove it with the limit comparsion test. I need to prove that it's ...
3
votes
1answer
50 views

$\sum_{n=0}^{\infty}\frac{a^2}{(1+a^2)^n}$ converges for all $a\in \mathbb{R}$

$$\sum_{n=0}^{\infty}\frac{a^2}{(1+a^2)^n}$$ Can I just see this series as a geometric series? Since $c = \frac{1}{1+a^2}<1$, we can see this as the geometric series: $$\sum_{n=0}^{\infty}bc^n = ...
1
vote
1answer
35 views

If $a_n\geq 0$ and $\sum_{n=1}^{\infty} a_n$ is convergent, then $\sum_{n=1}^{\infty} a_n^2$ and $\sum_{n=1}^{\infty} \frac{a_n}{a_n+1}$ converges

If $a_n\geq 0$ and $\sum_{n=1}^{\infty} a_n$ convergent, then $\sum_{n=1}^{\infty} a_n^2$ and $\sum_{n=1}^{\infty} \frac{a_n}{a_n+1}$ converges For $\sum a_n^2$ I used this to prove it like this: ...
0
votes
1answer
48 views

Uniform convergence of $\frac{\cos(nx)}{e^{nx}}$

We have a sequence $(f_n)$ on $[0,\infty)$, defined by $f_n(x)=\frac{\cos(nx)}{e^{(nx)}}$. The limit function $(f_n)$ of this sequence is $0$ for $x>0$ and $1$ for $x=0$. First part of the question ...
0
votes
0answers
19 views

These are the only two identities that yield triangle numbers, of this type $\sum_{n=1}^{m}(-1)^{n+m}n^k$. Am I right?

For $n:=1,2,3,...$ $T_n=\frac{n(n+1)}{2}:=1,3,6,10,...$ respectively These are the only two identities that yield triangle numbers, of this type $\sum_{n=1}^{m}(-1)^{n+m}n^k$. Am I right? (1) ...
0
votes
2answers
69 views

Proving that the series $\sum [\log(2n+1)-\log(2n)]$ diverges.

Let $f(n)=\log(2n+1)-\log(2n)$. Using the Cauchy's condensation test we have: $$2^nf(2^n)=2^n[\log(2\cdot2^n+1)-\log(2\cdot2^n)] = ...
4
votes
1answer
56 views

Check series for convergence

$$ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)^n}{n^{1/4}}\right) $$ I have no idea how to deal with it.
3
votes
2answers
49 views

Check for convergence

$$\sum_{n = 2}^\infty (-1)^n \sin\left( \frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n (\ln n)^2}\right)$$ I tried to use Maclaurin series, but failed to evaluate little-o.
0
votes
2answers
31 views

Convergent sequences must have bounded range?

I am currently reading Baby Rudin and I am having trouble understanding why convergent sequences must have a bounded range. Specifically, I am thinking of the following counterexample: $f(n)=1/(n-1)$ ...
0
votes
0answers
48 views

Series Convergence by Comparison [on hold]

I've been working on some calculus problems and am struggling to understand what I can compare this series to in order to prove convergence: $$ \sum_{n=0}^{\infty} \sin^2(\pi n) $$ I know that all ...
0
votes
0answers
12 views

Geometric or arithmetic sequences in polar coordinates

While studying the properties of the logarithmic spiral, I came across a theorem stating that secant lines drawn from the origin (pole) to the spiral form a geometric progression. I was interested to ...
3
votes
2answers
64 views

Behavior of a Collatz-like mod-4 sequence: Do some numbers increase without limit?

Define \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise} \end{eqnarray} and let $f^k(n) = f(f( \cdots (n) \cdots ...
1
vote
0answers
28 views

How can the integral of the sum of a geometric series apply for r=1

Say you have the series $R(x)=\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{2n+1}$, which is convergent for $x\in[-1,1]$ Then you differentiate: $R'(x)=\sum_{n=0}^{\infty}-(x^2)^n$ This is a geometric ...
0
votes
1answer
78 views

recursive generating functions

\begin{align} f(0) & = 1 \\ f(1) & = 1 \\ f(2) & = 2 \\ f(2n) & = f(n)+f(n+1), \;\;\;n\gt1 \\ f(2n+1) & = f(n-1)+f(n), \;\;\;n\ge1 \\ \end{align} I am trying to figure out ...
1
vote
0answers
20 views

Fix a typo involving the Lobachevsky function in Thurston's notes

I believe that there is a typo in these great notes Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, 1997), Chapter 7 that is provide us by MSRI, in the ...
4
votes
0answers
111 views

Prove $A=B=\pi$

$\gamma=0.57721566...$ $\phi=\frac{1+\sqrt5}{2}$ Let, ...
-4
votes
0answers
26 views

How do I solve these questions? [on hold]

A person receives Rs 6,000 per month salary.If his monthly salary is incremented by Rs 300 every year,what amount he would receive in 30 years? a 43,10,000 b 37,26,000 c 23,10,000 d 52,92,000 Q2 ...
3
votes
4answers
45 views

$a_n = \frac{1}{n}b_n$, $\lim b_n = L>0, L\in\mathbb{R}$, prove $\sum a_n$ diverges

I have to prove that if $$a_n = \frac{1}{n}b_n$$for $n\ge 1$ and $$\lim_{n\to\infty} b_n = L>0, L\in\mathbb{R}$$ then $$\sum_{n=1}^{\infty} a_n$$ diverges. My idea was to show that it's not true ...
2
votes
1answer
54 views

Solving $nx_n=(n+2)x_{n-1} + 1$ by the telescoping method

I am trying to solve this recurrence relation from a book "Problem solving through Problems" by Loren c. Larson (5.3.14 (b)) using the telescoping method. $$x_0=0\qquad nx_n=(n+2)x_{n-1} + 1\ (n > ...
2
votes
4answers
85 views

proof that $\sum_{n=1}^{\infty}\frac{n}{n^2+2n+1}$ diverges, by comparsion

I need to prove that $$\sum_{n=1}^{\infty}\frac{n}{n^2+2n+1}$$ diverges by comparsion. The way I did was to use $$\frac{n}{n^2+2n+1}>\frac{n}{n^2+2n^2+n^2} = \frac{n}{4n^2} = \frac{1}{4n}$$ ...
0
votes
1answer
24 views

Macularian series for natural log

So, I know that $$ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + ... $$ Am I right in assuming that I can derive to follow by a subtitution of $-x$ $$ln(1-x) = -x - \frac{x^2}{2} - ...
-2
votes
1answer
38 views

$\sum_{n=1}^{\infty}\frac{n^4}{n^4+1}$

$$n^4+n^4>n^4+1 \rightarrow \frac{1}{n^4+n^4}<\frac{1}{n^4+1}$$ for all $n>1$. Then: $$\frac{1}{n^4+n^4}<\frac{1}{n^4+1} \rightarrow \frac{n^4}{n^4+n^4}<\frac{n^4}{n^4+1} \rightarrow ...
5
votes
1answer
55 views

Asymptotic expansion of $(1+\epsilon)^{s/\epsilon}$

I have taken the logarithm of this expression and computed the Taylor expansion of the $\log(1+\epsilon)$ term but by doing this we're required to calculate powers of this series when using the ...
3
votes
4answers
63 views

Most natural way to prove $\sum_{n=1}^{\infty}\frac{1}{n+2}$ diverges

I don't know how my teacher wants me to prove that $$\sum_{n=1}^{\infty}\frac{1}{n+2}$$ diverges. All I know is that I have to use the $a_n>b_n$ criteria and prove that $b_n$ diverges. I tried ...
7
votes
1answer
144 views

Prove that sum is convergent

How to prove that the following sum is convergent? $$\sum_1^\infty\frac{\sin(n + \ln{n})}{n}$$ I tried to use formula $$\sin(n+ \ln{n}) = \sin{n}\cos \ln{n} + \sin \ln{n}\cos{n}$$ and $$\sum_1^N ...
1
vote
1answer
61 views

Prove $\frac{\gamma}{4}+\ln\left[\frac{\Gamma(1/4)}{4} \right]=\sum_{n=2}^{\infty}\frac{(-1)^n\zeta(n)}{2^{2n}n}$

The originate idea of this formula is from here (1) $$\frac{\gamma}{4}+\ln\left[\frac{\Gamma\left(\frac{1}{4}\right)}{4} \right]=\sum_{n=2}^{\infty}\frac{(-1)^n\zeta(n)}{2^{2n}n}$$ We arrived at ...
0
votes
0answers
17 views

check for relationship duplicate numbers

I 'm a programmer C# and look for one to formulate mathematical search numbers related to avoid duplicate relationships I need to make a relationship of users in the database and for that I would not ...
1
vote
1answer
37 views

Prove/disprove converge series

Can you help me or give me a hint with this, I don't know from where to start: prove/disprove this: $$\sum_{n=0}^\infty \frac{1}{(2n)!}=\frac{e^1+e^{-1}}{2}$$ Thanks!
0
votes
1answer
34 views

Is the sequence of functions $g_n=ng_1(nx)$ a Cauchy sequence?

Given a function $g_1(x) \in \mathcal L^2(\Bbb R)$ that satisfies: $$\int_{-\infty}^{\infty}dx \space g_1(x)=1$$ one can define a sequence of functions $g_n=ng_1(nx)$. Does $g_n(x)$ define a ...
-1
votes
2answers
49 views

Asymptotics of $\sum_{n}e^{-n^{2}}$.

Define the function $S(N)$ as $$S(N)=\sum_{n=0}^{N}e^{-n^{2}}$$ I am interested in the asymptotic behavior of $S(N)$ for large $N$. It is clear by the ratio test that $\lim_{N\rightarrow\infty}S(N)$ ...
1
vote
1answer
55 views

Show that the series converges uniformly

Show that $\sum_{n=1}^{\infty} \sin \left(\dfrac{x}{n}\right)^2$ converges uniformly on $[a,-a]$ , $a\in \mathbb{R}$. My attempt: Using Taylor's formula, we have:$$ \sin\left(\dfrac{x}{n}\right)^2 ...
0
votes
2answers
35 views

Recursive sequence nth element formula

What is the $n$th element of this sequence: $$S_n = S_{n-1} + (c_1 - S_{n-1})c_2$$ where $c_1$ and $c_2$ are constants and $S_1=0$. Thank you,
2
votes
2answers
56 views

Writing sequences using $\sum$ and $\prod$ symbols

Rewrite the following expressions using $\sum$ or $\prod$ a) $(x-1)(x-4)(x-9)(x-16)....(x-900)$ b) $1/(6^3) + 1/(9^4) + 1/(12^5) + 1/(15^6) +......+ 1/(33^{12})$ For part a) I noticed that ...
1
vote
1answer
31 views

Finding the values of $z$ such that $\sum_{n=0}^{\infty}(1+\sin{n})^nz^n$ converges

I'm trying to apply the nth root test to $$\sum_{n=0}^{\infty}(1+\sin{n})^nz^n$$ Hence I use that $\hat{R}=\left (\limsup |a_n|^{\frac{1}{n}}\right )^{-1}$ and get $$\hat{R}=\left (\limsup (1+\sin{n}) ...
1
vote
3answers
49 views

Finding a formula for a kth element in a sequence

I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$ The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - ...