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

learn more… | top users | synonyms (5)

3
votes
1answer
57 views

Rate of convergence for series.

1)What is convergence rate of a series \begin{equation} K(k) = {\frac{\pi}{2}} \hspace{1mm} {\sum_{m=0}^\infty}\binom{-1/2}{m}^2 k^{2m} \end{equation} Note that the presence of squares of ...
1
vote
2answers
47 views

Series, divergent

$\sum_{n=1}^\infty\frac{n^4-3n}{n^5+n}$ I want to show, that this series is divergent, but i can not find a divergent minorant... I tried ...
0
votes
2answers
51 views

What is the statistical difference (if any) between these two methods of generating an n-digit random number?

To preface, this question is coming from a software developer so it's written from that perspective. If I need to generate a random number with $n$ digits, I could do it in one of two ways. a. Ask a ...
1
vote
1answer
57 views

Determine the radius of convergence of the power series $\sum_{k=1}^\infty \frac{1-(-2)^{(-k-1)}k!}{k!} (z-2)^k$.

I could use some help solving this one. Applying the nth-root or the ratio test didn't work out for me. $$\sum_{k=1}^\infty \frac{1-(-2)^{(-k-1)}k!}{k!} (z-2)^k$$ Hints are just as appreciated ...
1
vote
1answer
48 views

Redundance in $l^p$ space.

I am covering the following problem: Let $A \subset l^p$, with $p \in [1,\infty)$, then is equivalent: i)A is relatively compact ii)A is bounded and we have $$\lim_{n \rightarrow \infty} \sup_{x ...
2
votes
1answer
64 views

$\sum_{k=0}^{n} \frac{1}{k!}\leq \left(1+\frac{1}{n}\right)^{n+\alpha} $

Find a value $\alpha\in\mathbb{N}$ such that \begin{equation} \sum_{k=0}^{n} \frac{1}{k!}\leq \left(1+\frac{1}{n}\right)^{n+\alpha} \end{equation} $\forall n\in\mathbb{N}$.
0
votes
1answer
34 views

Limit of $\frac{3^n\ +\ (-2)^n}{3^{n + 1}\ +\ (-1)^n}$

I'm trying to find out the limit of this sequence $\frac{3^{n}\ +\ \left(-2\right)^{n}}{3^{n\ +\ 1}\ +\ \left(-1\right)^{n}}$ I already know it is $\frac{1}{3}$. My idea was, to potentiate the whole ...
0
votes
3answers
119 views

Calculating sum of a series

$\sum_{n=1}^{\infty} nx^n$ for $x \neq 1$ It is quite obvious that for $q>1$ the sum will be $\infty$, but how to calculate it for $q<1$? Also, here is a solution with a derivative, but I want ...
1
vote
1answer
57 views

Sum to infinite terms faster method

Finding sum to infinite terms of series: $$\frac{1}{1\cdot3\cdot5} + \frac{1}{3\cdot5\cdot7} + \frac{1}{5\cdot7\cdot9} + \cdots$$ I approached this question by writing the general term first,then ...
3
votes
2answers
95 views

Limit and sum question [closed]

Solve these limits. $$\lim_{n\to\infty}\left(\,\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{2n}\,\right)=?$$ and $\lim_{n \to \infty}a_{n} = ?$ where $$a_{0}=1\,,\quad ...
2
votes
1answer
84 views

Approximating $\tanh(B\sqrt{A} )$ for small $A$ and arbitrary $B$ by correcting for asymptotic behaviour of $\tanh$

I would like to approximate a function containing terms of the form $\tanh( B\sqrt{A})$ for small $A$. I have tried doing a Taylor series, but I consistently find that it is not only $A$ that has to ...
0
votes
3answers
79 views

Is a sequence convergent and if so what is the sum

The sum $$\sum_{n=1}^{\infty} \frac{1}{(n+3)(n+2)} $$ Ive made it into partial fractions which gives $\frac{1}{n+2} - \frac{1}{n+3}$ But im unsure how to tell if this now converges as obviously as ...
0
votes
2answers
30 views

Determine the value of a series

The series is: $\sum_{n=1}^\infty 2n (-\frac{4}{9})^n x^{2n-1}$ I think I have to simplify it to have a cosinus and something else. I tried a few times but I just cant get it to: something + or * ...
1
vote
1answer
60 views

Why can the limit of a sequence approach a number and converge, but the limit of the series must approach $0$ to converge?

My question may not make much sense because I'm still trying to wrap my mind around infinite sequences and series. I seem to have good working knowledge of when and why to apply a certain tests for a ...
0
votes
1answer
17 views

Radius of convergence of a series (text problem)

the series: $\sum_{n=1}^\infty a_nx^n$ where $(a_n)_n$ is a limited sequence with $L((a_n)_n) \subseteq \mathbb{R}\backslash \{0\}$ My main problem is to get to something to work with. I dont know ...
1
vote
3answers
68 views

Why can law of limits not be used [closed]

Let $(a_n)$ be bounded (but not necessarily convergent) i.e there exists $M>0$ such that $|a_n| \leq M $ for all $n$ in natural numbers. Now consider $(b_n)$ with $b_n := \frac{a_n}{n}$ a) ...
2
votes
1answer
114 views

Convergence of $\sum\limits_{n=1}^\infty \frac{n!}{n^n} \times (5x)^n$

I have to check for which $x$ the series converges/diverges. $\sum\limits_{n=1}^\infty\frac{n!}{n^n} \times (5x)^n$ I know that for $|x| < \frac{1}{5}e$ it converges and for $|x| > ...
3
votes
4answers
122 views

Limit for $\lim _{n \to \infty}(n+2)^{2}\sin\frac{1}{n}$

Can't prove the limit $$\lim_{n \to \infty}(n+2)^{2}\sin\frac{1}{n}=\infty.$$ by definition it should start: Let $M>0$. There exists an $N>0$ for every $n>N$: ...
1
vote
0answers
54 views

Taylor's Formula and 'z' values

I have attached Taylor's Formula, an exercise problem from a section on Taylor polynomials, and the solution to this exercise. I understand part a, expanding $f$ using Taylor polynomials is the ...
3
votes
3answers
222 views

Infinite series of tangential circles?

I want to show that for all $n$ there is some collection of $n + 2$ circles such that two of the circles ($A$ and $B$) are tangential to each of the remaining $n$ circles (but not to each other) and ...
4
votes
1answer
72 views

Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
11
votes
2answers
264 views

Value of $\frac{1}{\sqrt{3}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{15}}+\cdots$ ($n$ terms)

Sum $$\frac{1}{\sqrt{3}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{11}}+\frac{1}{\sqrt{11}+\sqrt{15}}+\cdots \text{ ($n$ terms)}$$ I know how to use the telescoping series method when the terms are in ...
1
vote
1answer
86 views

Equivalent Definitions of Divergence

I am having a hard time wrapping my head around the equivalence of two definitions of diverging ($+ \infty$). The first definition, which I asked a previous question about, was purely negating the ...
1
vote
1answer
151 views

If two sequences converge in a metric space, the sequence of the distances converges to the distance of limits of the sequences.

Let $(p_n)$ and $(q_n)$ be sequences in the metric space $(X, d)$ and assume that $p_n \rightarrow p \in X$ and $q_n \rightarrow q \in X$. Prove that $d(p_n, q_n)$ converges to $d(p, q)$. Ok, so ...
2
votes
1answer
49 views

Hypergeometric series of sine function

Can someone please help me express this integral in terms of hypergeometric series? $$\int\frac{\sin⁡((2n-1)x)}{\sin⁡x}\text dx$$
8
votes
3answers
116 views

Sum $S(n,c) = \sum_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$

Consider the sum $$S(n,c) = \sum_{i=1}^{n-1}\dfrac{i}{ci+(n-i)}$$ where $0\le c\le 1$. When $c=0$, $S(n,c)$ grows asymptotically as $n\log n$. When $c=1$, $S(n,c)$ grows asymptotically as $n$. ...
6
votes
4answers
589 views

If $\sum_{n = 1}^\infty {{a_n}}$ converges, then is $\sum_{n = 1}^\infty (1+a_n)^{-1}$ a convergent series?

If $\sum\limits_{n = 1}^\infty {{a_n}}$ is convergent (with ${a_n} > 0$, $\forall n\in\mathbb{Z}$), then is $\sum\limits_{n = 1}^\infty {\left( {\frac{1}{{1 + {a_n}}}} \right)}$ is a convergent ...
3
votes
2answers
124 views

Prove that $\{s_n\}$ convergence implies $\{|s_n|\}$ convergence?

Prove that $\{s_n\}$ convergence implies $\{\left|s_n\right|\}$ convergence? Proof. Let $\{s_n\}$ be a sequence of real numbers. We show that $s_n \to s$ implies $|s_n| \to t$ for $s, t \in ...
0
votes
1answer
55 views

Is this sequence monotone?

I'm to determine whether a series converges or not and for that i need to know whether this $$\{a_n\}=\dfrac{1+nx}{\sqrt{n^2+n^6x^2}}$$ is an non-increasing sequence. I already tried to put ...
0
votes
1answer
446 views

compound interest with geometric series

Were studying geometric sequences in maths and this came up as one of the questions: A mortgage is taken out for 150000 and is repaid annually with 20000 installments. Interest is charged on the ...
2
votes
3answers
111 views

Limit of sequence. with Factorial

Can't find the limit of this sequence : $$\frac{3^n(2n)!}{n!(2n)^n}$$ tried to solve this using the ratio test buy failed... need little help
1
vote
0answers
36 views

Sequence with a contraction mapping among others

Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - ...
2
votes
3answers
65 views

How to show that $\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n^2}=\infty$

Let's say I need to find the limit: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n^2}$$ So I know that the limit is $\infty$, but I'm not sure how to show it in situations like this. ...
2
votes
2answers
167 views

Prove sequence converges uniformly

Suppose that the sequence ${\rm f_{j}}\left(x\right)$ on the interval $\left[0, 1\right]$ satisfies $$ \left\vert\,{\rm f_{j}}\left(t\right) - {\rm f_{j}}\left(s\right)\,\right\vert \leq ...
2
votes
3answers
74 views

How I can calculate $\sum_{n=0}^\infty{\frac{n}{3^n}}$? [duplicate]

For more than one series of any search function known not seem to find the sum of this series: $$\sum_{n=0}^\infty{\frac{n}{3^n}}$$ I've found that converges with the quotient criterion. Could you ...
2
votes
1answer
86 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
2
votes
2answers
95 views

Find the sum of the series $1/1 + 1/(1 + 2) + 1/(1 + 2 + 3) + … = \sum_{n = 1}^{\infty} \frac {1} {\sum_{j=1}^n j}$.

Find the sum of the series $1/1 + 1/(1 + 2) + 1/(1 + 2 + 3) + ... = \sum_{n = 1}^{\infty} \frac {1} {\sum_{j=1}^n j}$. I've been trying to find the sum of the series above by using an inequality of ...
4
votes
1answer
49 views

Calculation of $\sum_{k=0}^\infty \sin(x)^{ki}$

Is this formula correct? $$\sum_{k=0}^\infty \sin(x)^{ki}=\frac{i}{\sin(\ln(\sin(x)))-i\cos(\ln(\sin(x)))+i}$$ How is it possible to give a proof of this equality? Thanks.
2
votes
1answer
274 views

The $\sum\limits_{n = 1}^\infty {\frac{{\sin (\sqrt n )}}{{{n^{\frac{3}{2}}}}}}$ series is absolutely convergent?

$$\sum\limits_{n=1}^\infty\frac{\sin(\sqrt n)}{n^{\frac{3}{2}}}$$ Let ${a_n} = \frac{{\sin (\sqrt n )}}{{{n^{\frac{3}{2}}}}}$, then, using the criterion of quotient I must prove that $\mathop {\lim ...
2
votes
1answer
68 views

Generalization of some infinite series containing binomial coefficients

On a page here. There are some infinite series in the form $$\sum_{k=1}^{\infty}\frac{k^n}{\binom{2k}{k}}=\frac{a}{b}+\frac{c \pi}{d}$$ Where $n \in {[0,1,2,3,4...]}$ and for some natural numbers ...
2
votes
2answers
45 views

Finding $x$. The summation of the floor of the equation.

I would appreciate if somebody could help me with the following problem Q:Finding $x$. The summation of the floor of the equation. $$\sum_{i=1}^{2013}\left\lfloor\frac{x}{i!}\right\rfloor=1001$$
1
vote
2answers
34 views

Sequence of functions, question regarding notation.

I am trying to solve a problem which says: Let $S \subset \mathbb R^{\mathbb N}$ and let $\{f_n\}_{n \in \mathbb N}$ a sequence of functions $f_n:S \to \mathbb R$ that converges uniformly to a ...
2
votes
1answer
44 views

Having Trouble Understanding The Integral Test for Series.

Here is the definition from the textbook I am using. Let $\{a_n\}$ be a sequence of positive terms. Suppose that $a_n=f(n)$, where $f$ is a continuous, positive, decreasing function of $x$ for all ...
2
votes
2answers
89 views

Limit $\lim_{n\to\infty}n\frac{\sin\frac{1}{n}-\frac{1}{n}}{1+\frac{1}{n}}$

I need to find a limit of a sequence: $$\lim_{n\to\infty}n\frac{\sin\frac{1}{n}-\frac{1}{n}}{1+\frac{1}{n}}$$ I tried to divide numerator and denominator by n, but it didn't help, as the limit became ...
16
votes
1answer
228 views

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$

Let $$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$ where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as ...
1
vote
1answer
73 views

Six-story card castle problems with general term

First of all sorry if i am mistaking word translation, i am not a native English speaker. I got a problem which i partially solved. I must do calculations with a six-story card castle ...
1
vote
0answers
105 views

Expectation involving a maximum of a sequence of i.i.d. Gaussians

Let $X_1,\ldots,X_n$ be a sequence of i.i.d. standard Gaussian random variables. Denote the maximum of this sequence by $M_n$. I am interested in evaluating the following expectation: ...
3
votes
1answer
57 views

$(a_n)_{n\in\mathbb{N}}\in [0,1]^{\mathbb{N}}, \sum_{n=1}^{\infty}a_n=\infty\Rightarrow \prod_{n=1}^{\infty}(1-a_n)=0$

Consider a sequence $(a_n)_{\in\mathbb{N}}\in [0,1]^{\mathbb{N}}$ with $\sum\limits_{n=1}^{\infty}a_n=\infty$. Show that then $$ \prod_{n=1}^{\infty}(1-a_n)=0. $$ How can I show ...
16
votes
1answer
240 views

Does $ \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit?

The entire question is essentially given in the title: For $ n $ a positive integer, does the sequence $ x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert} $ have a limit as $ n \to \infty $? ...
3
votes
0answers
88 views

Is this Euler sum evaluatable?

I ran into an Euler sum that appears to be rather tough. Apparently, it does indeed have a closed form, so I assume it is doable. May be a false assumption, though. :) $\displaystyle ...