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

learn more… | top users | synonyms (5)

0
votes
0answers
23 views

Convergence of functional series $\sum_{k=1}^\infty x^k \tan(\frac{x}{2^k}) $

This is my functional series: $$\sum_{k=1}^\infty x^k \tan(\frac{x}{2^k}) $$ Now, to get the convergence radius (not sure if that's the correct word for it in English), I've been taught that I need ...
1
vote
1answer
42 views

|Determine N of series $\sum^N_{n=1}(-1)^n\frac{1}{2n+1}$ so that it differs from the actual sum by less than 0.001

Determine N of series $\sum^N_{n=1}(-1)^n\frac{1}{2n+1}$ so that it differs from $\sum^\infty_{n=1}(-1)^n\frac{1}{2n+1}$ by less than $\frac1{1000}$ How do I do this problem? I know that the ...
0
votes
2answers
29 views

Limit(s) of a sequence in the discrete topology

Why is the limit of the sequence $\{\frac{1}{n}: n \geq 1\}$ in the space $\mathbb{R}$ with the discrete topology the empty set? Intuitively the limit seems to be $\{0\}$ right?
0
votes
2answers
36 views

Split series to alternating series

Let $a_n$ be sequence of positive numbers. Is that true that: $\sum_{n=1}^{\infty} (-1)^n\cdot a_n$ converges $\implies$ $\sum_{n=1}^{\infty} a_{2n}-a_{2n+1} $ converges $\sum_{n=1}^{\infty} ...
4
votes
1answer
40 views

IEEE 754-like definition of “real” real numbers

once I have mapped, geometrically or by $\sin\left( \arctan\left( x \right) \right)$ the range $\left[ 0,+\infty \right[$ into $\left[ 0,1 \right[$, it's very "nice" to write (real) numbers in $\...
0
votes
1answer
28 views

Can we deduce that $0<a<ψ_{k}$ for infinitely many $k$?

Let $(ψ_{k})_{k≥1}$ be a bounded, positive and strictely decreasing real sequence ($ψ_{k}>ψ_{k+1}$), then it converges to its greatest lower bound $a>0$. My question is: Can we deduce that $$0&...
1
vote
1answer
36 views

Sequence of complex numbers $\frac{\tan(in)}{n}$. Is it convergent or divergent?

Consider the sequence of complex numbers $$z_n= \frac{\tan(in)}{n}$$ for all $n \in \mathbb N$. Is it convergent or divergent?
3
votes
2answers
107 views

How to prove this limit about $\gamma=\lim_{N\to \infty }\left(\sum_{n=1}^N\frac{1}{n}-\ln N\right)$

I have no idea how to prove it. $$\lim_{m\rightarrow \infty }\left [ -\frac{1}{2m}+\ln\left ( \frac{e}{m} \right )+\sum_{n=2}^{m}\left ( \frac{1}{n}-\frac{\zeta \left ( 1-n \right )}{m^{n}} \right ) \...
0
votes
2answers
29 views

Does the property of changing sign holds true for real sequences?

We know that if a function $f:ℝ→ℝ$ has infinitely many zeros, then it changes its sign infinitely many times. My question is: Does the same property holds true for real sequences $(u_{k})_{k≥1}$?
10
votes
5answers
200 views

Infinite summation: $x+x+x+x+… =2$?

One of my favourite little math problems is this $x^{x^{x^{x^{...}}}}=2$ The solution to it is quite simple. An infinite tower of x's is equal to 2, and above the first x there is still an infinite ...
2
votes
1answer
49 views

If $X$ is a normed space over $\mathbb{R}$ and $(x_{n})\rightharpoonup x\in X$ then $(x_{n})$ is bounded?

So if $X$ is a normed space over $\mathbb{R}$ and $(x_{n})$ is a weakly convergent sequence then I want to prove that $(x_{n})$ is bounded. Since $(x_{n})$ is weakly convergent we have that $f(x_{n})\...
3
votes
1answer
42 views

Proving an inequality of three sequences

Let two sequences ($u_n$) and ($v_n$): \begin{align} u_n &= \sin(1/n^2) + \sin(2/n^2) + \ldots + \sin(n/n^2)\\[0.2cm] v_n &= 1/n^2 + 2/n^2 + .... + n/n^2 \end{align} In the previous parts of ...
3
votes
2answers
93 views

Prove these identities using Jacobi's triple product identity.

I am requesting help with deriving some identities from Jacobi's triple product identity: $$\sum_{n=-\infty}^{\infty}z^nq^{n^2}=\prod_{n\geq 0}(1-q^{2n+2})(1+zq^{2n+1})(1+z^{-1}q^{2n+1})$$ Here is ...
0
votes
3answers
40 views

Is $\{x_n\}$ converges?

$\{x_n\}$ is a sequence of real numbers. For any two sub-sequences $\{u_n\}$ and $\{v_n\}$ (with no repeated terms) of $\{x_n\}$, $|u_n-v_n|\rightarrow 0$ as $n\rightarrow \infty$. Then can we ...
1
vote
2answers
51 views

Convergence of Infinite Sum [closed]

Using Mathematica, I have been able to make the following statement based on numerical evidence: $$\sum_{i=0}^\infty \frac{2^i}{x^i}=\frac{x}{x-2}$$ for any $x≥3$. How can this be proven?
0
votes
2answers
44 views

Prove series convergence of $\sum_{n=0}^{\infty} a^r_n$

Let $\{a_n\}$ be a sequence of non-negative real numbers. Prove that if $\sum_{n=0}^{\infty} a_n$ converges, then so does $\sum_{n=0}^{\infty} {a_n}^r$, where $r$ is a positive integer. I don't know ...
37
votes
1answer
690 views

Mirror algorithm for computing $\pi$ and $e$ - does it hint on some connection between them?

Benoit Cloitre offered two 'mirror sequences', which allow to compute $\pi$ and $e$ in similar ways: $$u_{n+2}=u_{n+1}+\frac{u_n}{n}$$ $$v_{n+2}=\frac{v_{n+1}}{n}+v_{n}$$ $$u_1=v_1=0$$ $$u_2=v_2=...
2
votes
0answers
42 views

Show an sequence that can not use l’Hôpital’s Rule to find limit

This from my textbook I understand that the domain of a function need to be real numbers to apply l’Hôpital’s Rule. But the textbook also provides a theorem below. It looks like to me that we can ...
1
vote
1answer
30 views

Which Test Is Appropriate to Determine Whether $\sum_{k=0}^{\infty} \frac{\mathrm{sin}\big(n+\frac{1}{2}\big) \cdot \pi}{1 + \sqrt{n}}$ Converges?

I wish to determine whether or not the following infinite series converges: $$\sum_{n=0}^{\infty} \frac{\mathrm{sin}\left(\big(n+\frac{1}{2}\big) \pi\right)}{1 + \sqrt{n}}$$ The problem is in the ...
1
vote
0answers
33 views

Performing Taylor series with 'fractional' powers

Taylor expansions are used all the time for physics problems, but sometimes they don't work because we expanded in the 'wrong' parameter. For example, suppose we're doing a relativistic kinematics ...
0
votes
1answer
26 views

Does an infinite series with an unbounded number of terms with the same value converge

Suppose one has an infinite series of positive reals. And suppose that there are an infinite number of sets of terms $S_n$ which take the same value. That is: $S_1 = \{ a_{1,1}, a_{1,2},...,a_{1,m}\}$...
0
votes
2answers
60 views

How can I prove this finite sum?

$$\sum_{i=0}^m \frac{i}{2^i} = 2-\frac{(m+2)}{2^m},\forall m \in \mathbb N$$
0
votes
1answer
24 views

Convergence of series given element wise convergence

We are given positive functions $f_n(t)$ with limits (as $t \rightarrow 0$) $\overline{f}_n$. Consider the following statements $f_n(t) \rightarrow \overline{f}_n$ as $t \rightarrow 0$ for all $n$. ...
0
votes
2answers
45 views

Proving that the sign of the sum of this alternating sequence depends on the sign of the largest number.

Listing the powers of 2 in order, i.e, "1, 2, 4, 8, 16,..." Then alternate their signs and add them. For example: -1 + 2 -4 + 8... or 1 - 2 + 4 - 8.... Prove that the sum will be positive if ...
0
votes
0answers
39 views

Show that the sums $\sum_{r \neq s} \frac{1}{\lambda_r -\lambda_s} - \sum_{r \neq t} \frac{1}{\lambda_r -\lambda_t}$ cancel out.

Suppose that $\lambda_r$ are distinct real numbers with $|\lambda_r - \lambda_s|\geq \delta$ if $r \neq s$. And let $z_r$ be any complex numbers. Then why does the sum $$ \mathop{\sum \sum}_{s \neq t} ...
0
votes
1answer
53 views

How to prove that a sequence is Cauchy in metric spaces [duplicate]

$(A,d)$ a metric space. Let $(a_n)_{n\geq 0}$ be a sequence such that there exists a $k<1$ and for all $n\geq 1$ we get $d(a_{n+1},a_n)\leq kd(a_n,a_{n-1})$. How do I prove that $(a_n)_{n\...
5
votes
2answers
1k views

The logic behind a sequence

I am trying to get the logic behind the sequence: for $n=2,3,\ldots$ $$\left(\frac{\log (2)}{\log \left(\frac{3}{2}\right)},\frac{\log (3)}{\log \left(\frac{17}{9}\right)},\frac{\log (4)}{\log \left(\...
1
vote
3answers
63 views

What is the computational benefit of Aitken's $\Delta^2$ process?

Let $(x_n)$ be a linearly convergent sequence. Then $$y_n := x_n - \frac{(x_{n+1}-x_n)^2}{x_{n-2} - 2x_{n+1} + x_n}$$ is called Aitken's $\Delta^2$ process. Remarkably, $(y_n)$ converges faster than $(...
4
votes
1answer
78 views

Uniformly continuous functions sequence $f_n(x)$ converges uniformly to a uniformly continuous function $f(x)$? [duplicate]

We know that if continuous functions sequence $g_n(x)$ converges uniformly to $g(x)$, then $g(x)$ is continuous function. But what if uniformly continuous functions sequence $f_n(x)$ converges ...
2
votes
1answer
36 views

How to judge the convergence of this series?

$$\sum_{n=1}^{\infty }\nu _{n}^{s}=\nu _{1}^{s}+\nu _{2}^{s}+\cdots +\nu _{n}^{s}+\cdots$$ where $\nu _{1}=\sin x>0~,~\nu _{n+1}=\sin \nu _{n}~,~n=1,2,\cdots$ I don't know how to do it.What 's' ...
0
votes
4answers
52 views

Suppose lim $p_n = p$ and lim $q_n = q$. If $p_n < q_n$ $\forall n \ge 1$, prove that $p \leq q$.

So i'm not sure how to approach this problem. By convergence, I know that $|p_n - p| < \epsilon$ and $|q_n - q| < \epsilon$, $\forall \epsilon$. But where do I go from here? I think I get the ...
2
votes
1answer
35 views

Sum of Sequentially Spaced Binomial Terms

Understanding that if $k>n$, we have that $\binom{n}{k}=0$, has there been any success coming up with closed formulas or asymptotic formulas for the following... $$B(n,k,j)=\sum_{i=0}^n\binom{n}{...
0
votes
1answer
23 views

Proving that if $x$ is a limit point of a sequence, then any interval with midpoint $x$ is a lure.

Let $x$ be a limit point of a sequence {${x_n}$}. Then for any given ϵ that is a positive number and any number $k$, there is an integer $n>k$ such that $|x_n-a|<ϵ$. Now let us consider the ...
0
votes
0answers
28 views

Multiple of angles trig formula using De Moivre's form: $f(n,k)=\cos(nk)$

I was wondering about other ways by which one could derive a formula for the multiplying of angles in a trig function when I (finally) understood how to do it using De Moivre's form: $$\cos(nk)=\Re(\...
0
votes
2answers
33 views

Simplifying Taylor Series for the function $\sqrt{2+x}$ about $x=3$

I have a past exam with the solution here. But I can't understand how they combined and simplified the terms going from the third last to the second last line. How did they get rid of $[1 \cdot 3 \...
0
votes
1answer
23 views

Proving that every trap is a lure

I am using the book "Learn Limits Through Problems". It states that an interval on a infinite sequence is a "trap" if a finite terms lie outside the interval while an interval will be called a "lure" ...
0
votes
1answer
70 views

Let's $1,2,3,\cdots,2005,2006,2007,2009,2012,2016,\cdots$ a sequence of integers defined by :

Let's $1,2,3,\cdots,2005,2006,2007,2009,2012,2016,\cdots$ a sequence of integers defined by : $ x_{k}=k$ if $1\leq k\leq 2006$ And $ x_{k+1}=x_{k}+x_{k-2005}$ if $k\geq 2006 $ Prove ...
-1
votes
1answer
21 views

Plus-Minus Infinite Series

I was kind of bored so decided to try and find a way to express ∓1±(1/2)∓(1/3)±(1/4)∓(1/5)... as an infinite sum with sigma notation. Failing dismally so far. Help?
0
votes
3answers
55 views

Detect five consecutive unsorted integers

It appears that the product of the differences between 3 consecutive integers in whatever order is always equal to 2. However, I can't find the pattern for 5 integers. We could sort them and get an ...
0
votes
2answers
66 views

Taylor series expansion of $ f(x)=e^{-x^2}$

How to find Taylor series expansion of $f(x)=e^{-x^2}$ What I'm stuck at is proving that the error $$R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}x^{n+1}$$ of the expansion tends to zero.
1
vote
2answers
57 views

How to solve $\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+…$

As my question says, $$\frac 1{1.2.3}+\frac 1{2.3.4}+\frac 1{3.4.5}+...$$ As far as $\frac 1{1.2}+\frac 1{2.3}+\frac 1{3.4}+...$ is concerned, we can write it as $\frac 1{n-1} - \frac 1{n}$ What ...
9
votes
2answers
145 views

$l^2$ convergence

How can I show the following known fact (fact not known to me) $$\text{let}\quad\{y_n\}_{n=1}^{\infty} \in l^2\quad\text{then}\quad\left\{ \frac{1}{n}\sum_{j=1}^{n} y_j\right\}_{n=1}^{\infty} \in l^2\...
4
votes
0answers
100 views

Finite differences of power functions

I'm interested in finite differences, to be precise, finite differences $\Delta^n f(x)$, where $n \in \mathbb{N}$ and $f$ is a real function given by $f(x) = x^a$ for some $a \in \mathbb{R}$. I use ...
1
vote
2answers
52 views

$\sum_1^\infty e^{n(1-x)}x^n$ and it's Uniform/Pointwise Convergence on different intervals

Find largest subset $X \subset [0,\infty)$ on which $\sum_1^\infty e^{n(1-x)}x^n$ is convergent: Find the region where $\sum_1^\infty e^{n(1-x)}x^n<\infty$. Find $n+1$ term and the $n$ terms. $...
5
votes
3answers
108 views

Finding the sum to n terms of series :$\frac{1}{1\cdot 2\cdot 3\cdot 4} +\frac{1}{2\cdot 3\cdot 4\cdot 5} + \frac{1}{3\cdot 4\cdot 5\cdot 6}+\cdots$

$$ \frac{1}{1\cdot 2\cdot 3\cdot 4} +\frac{1}{2\cdot 3\cdot 4\cdot 5} + \frac{1}{3\cdot 4\cdot 5\cdot 6}+\cdots $$ up to $n$ terms. I need help in solving this sum. I tried finding the coefficients of ...
0
votes
3answers
129 views

How to fit $\sum{n^{2}x^{n}}$ into a generating function?

I do have somewhat of a reasoning for this. $$S = 1 + x + x^{2} + x^{3} +.. + x^{n} $$ Denoting the derivative of $S$ as $T$ $$T = 1 + 2x + 3x^{2} + 4x^{3} +... + nx^{n-1}$$ $$xT = x + 2x^{2} + ...
1
vote
3answers
116 views

Why does this series diverge?

Let $a_n=\frac{4^n n!}{n^n}$ be a series. Why does it diverge? I got that $a_n\le a_{n+1}$, but $n^n$ is rising much faster than $4^n$ and $n!$?
3
votes
0answers
38 views

convergence test of $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^2+4n+5} $

$$\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^2+4n+5} $$ Can I say that $\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^2+4n+5}\leq\sum_{n=1}^{\infty} \frac{\sqrt{n}}{n^2}=\sum_{n=1}^{\infty} \frac{1}{n^{\frac{3}{...
0
votes
6answers
92 views

Prove that the sequence: $a_1 = 1, a_{n+1} =\sqrt{c+da_n}$ (when the real numbers $c, d > 1$) is converging and find it's limit

I have a summarized solution but it's starts with proving that the sequence is bounded from above by c+d. How can I know that this sequence is bounded by c+d? I understand the proof by induction but ...
0
votes
1answer
37 views

of the following four statements which are true?

let $f$ be a continuously differentiable function on $\mathbb R$. let $f_n(x)$= $n(f(x+\frac{1}{n})-f(x))$ then 1) $f_n$ converges uniformly on $\mathbb R$ 2) $f_n$ converges on $\mathbb R$ but ...