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

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3
votes
1answer
160 views

Reducing a power series to a rational function

I have the series $$\sum_{n=0}^\infty (-1)^n 2n x^{(2n -1)}$$ It turns out that this series is equal to the function $$\frac{-2x}{(1+x^2)^2}$$ Is there a general method that would demonstrate this ...
1
vote
2answers
69 views

convergence and limits

How can I rewrite $\frac{1}{1+nx}$ and prove it's absolute convergence as $n \rightarrow \infty$? Given $\epsilon > 0$, should I define $f_n(x)$ and $f(x)$? Any help is hugely appreciated. Thank ...
2
votes
5answers
855 views

How do I prove the sequence $x_{n+1}=\sqrt{\alpha +x_n}, x_0=\sqrt \alpha, \alpha>0$ converges? ( boundedness?)

I need to "study the limit behavior of the following sequence" and then compute the limit. I can compute the limit and prove the monotonicity, but I am having trouble proving boundedness. I tried to ...
0
votes
2answers
47 views

Verifying a Limit of a Sequence Using Integrals

Is there an alternative way of verifying this limit using integrals?
3
votes
3answers
108 views

Find $ \lim_{n \to \infty } z_n $ if $z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right) $

How do I approach the problem? Q: Let $ \displaystyle z_{n+1} = \frac{1}{2} \left( z_n + \frac{1}{z_n} \right)$ where $ n = 0, 1, 2, \ldots $ and $\frac{-\pi}{2} < \arg (z_0) < ...
11
votes
3answers
311 views

Closed form for $\sum_{k=0}^{n} k\binom{n}{k}\log\binom{n}{k}$

Is it possible to write this in closed form: $$\sum_{k=0}^{n} k\binom{n}{k}\log\left(\vphantom{\Huge A}\binom{n}{k}\right)$$ Can you get something like $$n2^{n-1}\log(2^{n-1})$$
1
vote
2answers
389 views

Convergent sequences and proof

Prove that $\dfrac{1+n}{n^2}$ converges as $n \to \infty$ How do I go about constructing this proof? Can I use the definition that $\operatorname{abs}(a_n - L < \epsilon)$?
2
votes
1answer
327 views

A convergence problem: splitting a double sum

I have been facing some difficulties with the following question. For an absolutely convergent series $\sum_m a_m$, and the Möbius function $\mu(n)$, $x=(x_1,x_2)\in \mathbb{R}^2$, and $\alpha ...
0
votes
1answer
50 views

Weak and normwise convergence of sequence of linear functionals

Is this sequence of linear functionals weakly (normwise) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_1\,?$$
4
votes
4answers
2k views

Is there a rule for prime numbers?

I've passed by this article: http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/ and this paper: http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf They ...
2
votes
2answers
314 views

Limit sequence proof

Let $X=(x_n)$ be a sequence of strictly positive real numbers such that $\lim\left(\frac{x_{n+1}}{x_n}\right)<1$. Show that for some $r$ with $0<r<1$ and some $C>0$, then we have ...
6
votes
3answers
121 views

Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$

Does anybody know how to prove this series? $$\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$$ I arrived at this through Mathematica. I tried writing ...
1
vote
1answer
334 views

How to show this sequence of functions weakly converge?

$X = C[0,1]$. $$x_n(t) = \begin{cases} nt, & \text{for $0 \leq t \leq \frac{1}{n}$ } \\ 2-nt, & \text{for $\frac{1}{n} \leq t \leq \frac{2}{n}$ } \\ 0, & \text{for $\frac{2}{n} \leq t ...
8
votes
1answer
85 views

Proving $\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$?

$$\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$$ $P$ is primes. Interesting question ran across while tutoring. ...
1
vote
2answers
107 views

How do I compute the limit of this sequence $x_{n+1}=\frac{(x_n)^2+b}{2}$, $x_o=0$ and $b\in [0,1]$

I need to study the limit behavior of the sequence and discuss all possible situations with the parameters given. The sequence is $x_{n+1}=\frac{(x_n)^2+b}{2}$, $x_o=0$ and $b\in [0,1]$. This can ...
0
votes
3answers
142 views

A sequence is defined by: $a(1) = 1$ and $a(n + 1) = 3a(n) - 1$ for $n \ge 1$. What is $a(100)$?

A sequence is defined by: $a(1) = 1$ and $a(n + 1) = 3a(n) - 1$ for $n \ge 1$. What is $a(100)$? Solution. The first few terms of $a(n)$ are $1,2,5,14,\ldots$. The general solution to the ...
5
votes
5answers
1k views

Function of $ \sqrt{2+\sqrt{2+\sqrt{2+}}}\ldots $

$$\sqrt{2+\sqrt{2+\sqrt{2+}}}\ldots$$ How you could put this into a summation equation? I'm stuck. At one point thought it was this: $$ \sum_{n=0}^{x} 2^{\frac{1}{2^x}} $$ but that would just be ...
2
votes
3answers
101 views

Did I compute the limit of of the sequence $x_{n+1}=\frac{x_n}{x_n+1}, x_o=1$ properly?

I need to study the limit behavior of $x_{n+1}=\frac{x_n}{x_n+1}, x_o=1$ and if the limit exists, compute the limit. I observed the first few terms and it seemed that the sequence was decreasing so I ...
1
vote
2answers
736 views

How to rigorously prove the convergence of an iterative sequence

Suppose we have an iterative sequence defined by $x_{n+1} = g(x_n)$ where $$g(x)= \frac{x^4 + 1}{3}$$ and we are looking at the two cases: $x_1 = 0$ $x_1 = 1$ While I know that if $x_n ...
3
votes
3answers
912 views

How do I find the limit of the sequence $a_n=\frac{n\cos(n)}{n^2+1}$ and prove it is a Cauchy sequence?

I need to study the limit behavior of $a_n=\frac{n\cos(n)}{n^2+1}$, which can be written as $\frac{n}{n^2+1}\cos(n).$ I knew that it wasn't going to be monotone because $cos(n)$ oscillates between -1 ...
5
votes
0answers
64 views

Functional sequence [duplicate]

Let $(f_n)$ be a sequence of functions $\mathbb{R} \rightarrow \mathbb{R}$. Suppose that for any $(x_n)$ convergent to $x$ we have $f_n(x_n) \rightarrow f(x)$. Prove that $f$ in continuous, there is ...
19
votes
2answers
760 views

On the set of the sub-sums of a given series

Choose a sequence $(x_n)_{n\in\mathbb N}$ of nonnegative real numbers with finite sum $x=\sum\limits_{n\in\mathbb N}x_n$ and consider the set $X=\{x_I\mid I\subseteq \mathbb N\}$ where, for every ...
2
votes
3answers
219 views

$ \sum_{n=2}^\infty \frac{1}{n^3(n^3+1)}. $

The series is: $$ \sum_{n=2}^\infty \frac{1}{n^3(n^3+1)}. $$ I tried splitting the whole thing into simple fractions but I don't seem to get anywhere. Any ideas?
0
votes
0answers
37 views

Interesting D.E turns into Geometric Limit and Cyclic Group.

Me and a friend Observed some very odd behavior of a matrix when trying to solve an ODE Consider first the 3x3 system where a,b are parameters in the Reals and fine the general solution. ...
2
votes
3answers
254 views

How do I prove $\frac{n}{\sqrt[n]{n!}}$ is either monotonic and bounded, or Cauchy?

I need to "study the limit behavior of the following sequence" and then compute the limit. The sequence is $a_n=\frac{n}{\sqrt[n]{n!}} $. This can also be written as $(\frac{n^n}{n!})^{(1/n)}$ I tried ...
0
votes
1answer
259 views

Showing convergence for certain values of P

Determine for which values of p the following series converges: $$\sum_{n=2}^{\infty} (-1)^{n-1} \frac{(\ln n)^P}{n}$$ So far, just from looking at various values for $P$, it seems to be that any ...
1
vote
0answers
115 views

What does it mean for a series to be “about a” or “centered at a”. and what about polynomial approximations?

1) What does it mean for a series to be "about a" or "centered at a". 2) I also do not understand what does it mean for representing power series as sigma C (subscript n) times (x-a) to the power of ...
47
votes
6answers
3k views

How to prove this identity $\pi=\sum\limits_{k=-\infty}^{\infty}\left(\frac{\sin(k)}{k}\right)^{2}\;$?

How to prove this identity? $$\pi=\sum_{k=-\infty}^{\infty}\left(\dfrac{\sin(k)}{k}\right)^{2}\;$$ I found the above interesting identity in the book $\bf \pi$ Unleashed. Does anyone knows how to ...
0
votes
5answers
93 views

Is this a valid proof to show $\frac{n^5}{2^n}$ diverges?

Claim: $a_n = \frac{n^5}{2^n}$ diverges Let M be arbitrary Then $$ \forall \; n \ge \text{max} \big\{ \big[ \sqrt[3]{M} \, \big] + 1 , 3 \big\} \\ n > \sqrt[3]{M} \\ \implies n^3 > M $$ And ...
2
votes
3answers
97 views

How to find the solutions of this equation?

How to find all the solutions of the following equation? $1+\frac{x}{2!}+\frac{{{x}^{2}}}{4!}+\frac{{{x}^{3}}}{6!}+\frac{{{x}^{4}}}{8!}+...=0$
1
vote
1answer
413 views

Convergence radius of power series

I am trying to solve an exercise, but i am not sure that the result i get at the end is correct...May i kindly ask you for a little help or a remark? Find the radius of convergence of the following ...
0
votes
2answers
565 views

Strange Sequence

On a recent math test, I was challenged by the following question. What are the next three terms in this sequence: 5, 12, 10, 10, 16, 13....? Hint: We were ...
6
votes
1answer
304 views

Missing term in series expansion

I asked a similar question before, but now I can formulate it more concretely. I am trying to perform an expansion of the function $$f(x) = \sum_{n=1}^{\infty} \frac{K_2(nx)}{n^2 x^2},$$ for $x \ll ...
4
votes
3answers
976 views

How can I tell that the sequence $a_n=\frac {\ln(n)} {n}$ converges and to what it converges?

I need to "study the limit behavior" and compute the limit if it exists. This is what I have done so far. In order to study the limit behavior I tried to first check the monotonicity and boundedness ...
3
votes
1answer
210 views

A summation of a series with power, exponential and factorial

$\sum_{k=0}^\infty {\frac{k^{C_1} {C_2}^k}{k!}}$ where $C_1$ is a positive integer and $C_2$ is a real number. Is there a close form or an approximation of the result when the summation converges?
5
votes
7answers
219 views

How can I tell if the sequence $a_n=\frac {2^n} {n!}$ converges, and if it converges what is its limit?

I know this question was asked before, but I couldn't find it and I think my question is slightly different. I'm new here so if I do something wrong, I'm sorry, I'm still trying to learn. I need to ...
0
votes
0answers
151 views

Convergence of Nested Series

My book states that Pringsheim has proved that the following statement is true for certain conditions: ...
8
votes
1answer
100 views

Does $\sum_{k=1}^{\infty} \frac{1}{8^k+2^k+1}$ have a closed form?

Is there a closed form for this infinite summation? $$\sum_{k=1}^{\infty} \frac{1}{8^k+2^k+1}$$
6
votes
1answer
96 views

Non-induction proof for $\frac{1}{2\sqrt{2!}}+\frac{1}{3\sqrt[3]{3!}}+\cdots+\frac{1}{n\sqrt[n]{n!}}>\frac{n-1}{n+1}$

I know to prove it by induction but is this the only way ? $$\frac{1}{2\sqrt{2!}}+\frac{1}{3\sqrt[3]{3!}}+\cdots+\frac{1}{n\sqrt[n]{n!}}>\frac{n-1}{n+1}, \space n\ge2$$
0
votes
1answer
43 views

Is $3^{2n}$ geometric, if so find common ratio and sum or the first n terms?

$3^{2n}$ is it arithmetic or geometric, or neither. If arithmetic find common difference and sum of first n terms. If geometric, find common ratio and sum of first n terms.
1
vote
1answer
29 views

Simplest way to solve n in a series in which $n = 4\%$ of $ n-1$

I'm trying to figure out the best way to approach this simple problem. I have a series of numbers in which $n =4%$ of $n-1$ basically giving $100~4~0.16~0.0064~...$ The function should take three ...
8
votes
4answers
665 views

limit of an integral of a sequence of functions

Suppose that $f$ is continuous on $[0,1]$. ($f'(x)$ may or may not exist). How can I show that $$\lim_{n\rightarrow\infty} \int\limits_0^1 \frac{nf(x)}{1+n^2x^2} dx = \frac{\pi}{2}f(0)\;?$$ My ...
2
votes
3answers
236 views

Convergent Sequence with Limit Superior

I have been working ahead in my textbook, and came across this question at the end of the questions of the chapter, but there is no explanation of limit superior in the chapter. If someone could ...
5
votes
0answers
74 views

Convergence of $\sum^n_{k=1}\frac1k$ after removing terms containing the digit $p$ [duplicate]

We know that $\sum^n_{k=1}\frac1k$ diverges. But if I were to pick a digit $p$ so that $p$ is an integer between $0$ and $9$ inclusive, and then I removed all terms in the sum $\sum^n_{k=1}\frac1k$ ...
1
vote
1answer
67 views

What kind of geometric-like summation is this, and does it have a solution?

Thank you, I was running investment numbers when the following summation came up: $$ \sum_{n=0}^N(1+i_1)^n(1+i_2)^{N-n} $$ Does this have a closed solution like the geometric series does? I haven't ...
6
votes
1answer
70 views

question about limit and series

consider following hypotheses $ m\in\mathbb N$ $c\in \mathbb C\,$ ,$\, \; a_j\in \mathbb C$ $a_j\in \mathbb C\;$ , $\;|a_j|=1,\;\forall\;1\le j\le m$ if ...
9
votes
1answer
114 views

Evaluating a slow sum

In my integration adventures, I came across this sum which I could not simplify: $$\sum_{n=1}^{\infty}\frac{(-1)^{n}\log(2n+1)}{2n+1}$$ Wolfram seems to believe the sum diverges and is not of much ...
0
votes
1answer
74 views

Two quick questions about convergence (in the context of pointwise vs. uniform convergence)

I found this example online: Let $\{f_{n}\}$ be the sequence of functions on $(0,\infty)$ defined by $f_{n}(x)=\frac{nx}{1+n^{2}x^{2}}$ .This function converges pointwise to zero. ...
0
votes
1answer
106 views

Bounded recursive sequence

I would like to know if there are known bounded recursive sequence (non monotonic): It shouldn't be a constant, neither a convergent sequence, nor a periodic one. (I am not asking for a true random ...
0
votes
1answer
26 views

Rewriting the series $\sum_{k=0}^{\infty} \sum_{s=0}^{\infty} h(k) g(s) z^{-(k+s)}$ in terms of $l = k + s$ and $k$

In one of the text book I found the following expression $$\sum_{k=0}^{\infty} \sum_{s=0}^{\infty} h(k) g(s) z^{-(k+s)}$$ Letting $k+s=l$, then the book has written the following ...