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

learn more… | top users | synonyms (3)

17
votes
4answers
391 views

A double sum $\sum \limits_{n=1}^{n=\infty}\left(\sum \limits_{k=n}^{k=n^2}\frac{1}{k^2}\right)$

How to evaluate $\displaystyle\sum_{n=1}^{n=\infty}\left(\sum_{k=n}^{k=n^2}\frac{1}{k^2}\right)$?
2
votes
4answers
126 views

Sequence of numbers with infinite number of primes

If I have an infinite sequence of positive integers with infinite number of primes and if I have an infinite number of distinct sequences with such properties may I claim that there is an infinite ...
5
votes
3answers
623 views

Asymptotics of terms and errors in Stirling's Approximation

I have two related questions. Both are related to the asymptotics of Stirling's approximation, which is why I have included them in the same question. I will separate the questions if it is deemed ...
11
votes
3answers
941 views

closed form of $\sum \frac{1}{z^3 - n^3}$

I am currently trying to find a closed form expression for $\displaystyle f(z) = \sum_{n \in \mathbb{Z}} \frac{1}{z^3 - n^3}$, $z \in \mathbb{C}$. After a set of twists and turns, I have hit a wall. ...
10
votes
2answers
200 views

Is there an alternating series that satisfies only one of the conditions of the Alternating Series Test that nonetheless converges?

I was recently helping a college math student with her homework. Her teacher had offered an extra-credit question: Find two alternating series $\sum_{n=1}^\infty (-1)^{n-1}a_n$ such that $a_{n+1} ...
2
votes
3answers
389 views

limit superior of a sequence - showing an alternate definition

I am wondering if anybody can help me with a problem regarding the definition of the limit superior of a sequence - or rather showing an alternate but equivalent defintion holds. The question is: The ...
2
votes
2answers
88 views

Recursion relation

How does one show that $I_n=\int\limits_0^1 (1-x^2)^ndx $ satisfies the recursion relation $I_n={2n\over 2n+1}I_{n-1}$? I don't think I have to explicitly evaluate the integral right? Thanks.
4
votes
0answers
426 views

Convergent sum with primes

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
10
votes
2answers
199 views

$1+1/(2+3)+1/(4+5+6)…$

What is the value of the sum of the series $$\frac{1}{1}+\frac{1}{2+3}+\frac{1}{4+5+6}+\dotso\;?$$ And this: $$\frac{1}{1}+\frac{1}{2\cdot3}+\frac{1}{4\cdot5\cdot6}+\dotso\;?$$
3
votes
2answers
628 views

Using Taylor series expansion as a bound

I have a function $f(x)$ that has convergent Taylor series expansion around $x=0$ in the following form: ...
5
votes
3answers
284 views

Sum of the series $\sum_{n=1}^\infty \frac{(-1)^n}{n2^{n+1}}$

How do I calculate the sum of this series (studying for a test, not homework)? $$\sum_{n=1}^\infty \frac{(-1)^n}{n2^{n+1}}$$
2
votes
0answers
176 views

Can we confirm $\sum_x{x!}$?

According to Wikipedia's article on indefinite sums, they list the following formula near the bottom of the page: $$\displaystyle \sum_x{\Gamma(x)}=(-1)^{x+1}\Gamma(x)\frac{\Gamma(1-x,-1)}{e}+C$$ ...
2
votes
1answer
100 views

Two harmonic subseries

Let $S_0=1$, and let $S_{n+1}$ be the sum of $1/k+1/(k+1)+1/(k+2)\cdots 1/(k+n)$, for the integer $k$ such that $S_{n+1}$ is maximal while $S_{n+1}<S_n$. Does $\sum_{k=1}^{\infty}S_k(-1)^{k}$ ...
9
votes
2answers
159 views

Sum of a rearranged alternating harmonic series, with three positive terms followed by one negative term

The series is this: $$ 1 + 1/3 + 1/5 - 1/2 + 1/7 + 1/9 + 1/11 - 1/4 + 1/13 + 1/15 + 1/17 - 1/6 ...$$ The hint is to consider partial sums to $4n$ terms. I did that, and got the summation $$ \sum ...
7
votes
1answer
144 views

Question dealing with a series of functions that uniformly converges on $[0,1]$

Let $f_1, f_2, \ldots$ be a sequence of continuous positive functions of $[0,1]$ and let $a_n = \sup\{ f_n(x) : x \in [0,1]\}$. In class, we showed that if $\sum f_n$ uniformly converges on $[0,1]$, ...
5
votes
1answer
104 views

Convergence of a sequence, $a_n=\sum_1^nn/(n^2+k)$

Let $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}$ . I would like to know whether the given sequence converges. I see that, $ a_{n} = \sum_{k=1}^{n} \frac{n}{n^{2}+k}= \sum_{k=1}^{n} ...
6
votes
5answers
301 views

Limit proof of a sqrt-heavy expression with binomial formula / sandwich-rule

Let be $$b_n := \sqrt{n+\sqrt{2n}}-\sqrt{n-\sqrt{2n}}, n\in\mathbb{N}$$ a sequence. I am to determine $\lim\limits_{n\to\infty}b_n$ which is obviously $\sqrt{2}$. My first step was to transform the ...
0
votes
2answers
2k views

Sum of two conditionally convergent series

Suppose I have two conditionally convergent series $\sum \limits_{n=1}^{\infty} s_n$ and $\sum \limits_{n=1}^{\infty} t_n$. According to http://mathworld.wolfram.com/ConvergentSeries.html the series ...
13
votes
5answers
426 views

How do we show the equality of these two summations?

How do you show the following? $$\sum \limits_{i=1}^{n}\ \sum \limits_{j=i}^{n}\ \sum \limits_{k=i}^{j}\ 1 = \sum \limits_{j=1}^{n}\ \sum \limits_{k=1}^{j}\ \sum \limits_{i=1}^{k}\ 1 $$ It's not ...
2
votes
2answers
155 views

Proof for convergence of a given progression $a_n := n^n / n!$

"Examine whether the given progressions converge (eventually improperly) and determine their limits where applicable. (a) $$(a_n)_{n\in\mathbb{N}}:=\frac{n^n}{n!}$$ [...]" I am having problems ...
43
votes
4answers
2k views

How to sum this series for $\pi/2$ directly?

The sum of the series $$ \frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1} $$ can be derived by accelerating the Gregory Series $$ \frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2} ...
1
vote
1answer
105 views

Are there times when convergence tests contradict each other?

In my Calc II class, we're just starting convergence tests and all the examples are very convinient and they work perfectly (obviously, since they are examples), but my professor couldn't really ...
3
votes
1answer
76 views

Prove that all but finite number of members of sequence are positive if limit is positive

Prove that if $x > 0$ and $x_n$ is a sequence with $\lim\limits_{n \to \infty} x_n = x$, then there is a real number $N$ s.t. whenever $n > N$, $x_n > 0$. This is a homework question and I'm ...
7
votes
1answer
184 views

Sum of reciprocals of primes

If $p_i$ is an infinite set of distinct primes such that $c=\sum\frac{1}{p_i} < \infty$, must $c$ be a transcendental number?
6
votes
1answer
242 views

Probabilistic proof of existence of an integer

The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$. Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...
5
votes
3answers
121 views

Transcendentals that sum to rationals

Is there any sequence $a_n$ of transcendental numbers, such that $ma_i\neq na_j$ for all integers m,n,i,j and any partial sum of $a_k$ is transcendental, but the total sum is rational ?
3
votes
3answers
514 views

Strong Mathematical Induction Recursion Inequality

I have a question that is for a homework assignment and I just would like to ask if I seem to be on the right track or if I'm just doing it completely wrong. Here is the question: The sequence ...
8
votes
3answers
187 views

For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?

The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges. What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
6
votes
1answer
2k views

L'Hopital's rule and series convergence

I teach freshman calculus, and have recently been discussing series. In one question on a recent test, I asked whether $\sum\frac{n^2}{n^3+1}$ converges or diverges. One student got the correct ...
2
votes
2answers
247 views

Studying the series $ \sum \prod\limits_{k=2}^n \left(1+\frac{(-1)^k}{\sqrt{k}} \right) $

In order to study the series $\sum u_n$ where $$u_n=\prod_{k=2}^n \left(1+\frac{(-1)^k}{\sqrt{k}} \right), $$ I'm trying to express $$\ln u_n= \sum_{k=2}^n \ln \left(1+\frac{(-1)^k}{\sqrt{k}} ...
1
vote
2answers
126 views

Does this series converge?

I have the following series as an exercise but for some reason i cannot prove if it converges or not. I used the integral test. The series is positive and decreasing so we can use it( i will not ...
1
vote
3answers
125 views

Convergence of a positive series

Let $$\begin{align*} \sum u_{n},\sum v_{n} \end{align*}$$ be two positive convergent series. How can I prove that $$\begin{align*} \sum \frac{u_{n}v_{n}}{au_n+bv_n}; a,b>0 \end{align*}$$ ...
9
votes
2answers
326 views

Find $ \lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}{\frac{2n}{(n+2i)^2}} $

Find $$ \lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}{\frac{2n}{(n+2i)^2}}.$$ I have tried dividing through by $1/n^2$ and various other algebraic tricks but cannot seem to make any progress on this ...
3
votes
2answers
132 views

How to simplify $f(x)=\sum\limits_{i=0}^{\infty}\frac{x^{i \;\bmod (k-1)}}{i!}$?

$$f(x)=\sum_{i=0}^{\infty}\frac{x^{i \;\bmod (k-1)}}{i!}$$ ${i \bmod (k-1)}$ $\quad$ says the $x$ powers can be only $x^0$, $x^1$, ...,$x^{k-2}$ Understand simplify a way to transform this infinity ...
1
vote
1answer
266 views

Maclaurin series of $\frac{1}{1+x^2}$

I'm stumped here. I''m supposed to find the Maclaurin series of $\frac1{1+x^2}$, but I'm not sure what to do. I know the general idea: find $\displaystyle\sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!}x^n$. ...
1
vote
2answers
309 views

Any converging sequence is bounded

Any converging sequence is bounded. This is a theorem in our book. It is proved by picking the first (finite) number of elements and show that the others are smaller than some constant. But, I ...
2
votes
1answer
698 views

Supremum of a sequence of function

Let $(f_n)_{n \in \mathbb{N}}$ a sequence of function from a nonempty-set $X$ to $\bar{\mathbb{R}}=[-\infty,\infty]$, $g\colon X \to \bar{\mathbb{R}}$ defined by $g(x)=\sup_{n \in \mathbb{N}} f_n ...
1
vote
2answers
169 views

Finding a harmonic number closest to a given large integer

So, today in my real analysis course, a theorem was given: $$\sum a_n \text{ converges }\iff \{s_n\}_{n=1}^{\infty} \text{ is bounded,}$$ where $s_n$ is the $n$-th partial sum of $\sum a_n.$ Now, of ...
2
votes
4answers
166 views

Simplify an expression to show equivalence

I am trying to simplify the following expression I have encountered in a book $\sum_{k=0}^{K-1}\left(\begin{array}{c} K\\ k+1 \end{array}\right)x^{k+1}(1-x)^{K-1-k}$ and according to the book, it ...
4
votes
4answers
1k views

Prove sequence $a_n=n^{1/n}$ is convergent

How to prove that the sequence $a_n=n^{1/n}$ is convergent using definition of convergence?
1
vote
1answer
189 views

Pi approximation

If $d(a,b)=$ largest $n$ such that $a$ and $b$ agree on all digits upto $n$. Eg. $d(\pi,3.14)=3$, $d(0.1234667,0.1234669)=7$. What is the asymptotics of $d(\pi/4,1-1/3+1/5-1/7+\cdots(\pm)1/m)$ as ...
4
votes
3answers
445 views

What would be the radius of convergence of $\sum\limits_{n=0}^{\infty} z^{3^n}$?

I know how to find the radius of convergence of a power series $\sum\limits_{n=0}^{\infty} a_nz^n$, but how does this apply to the power series $\sum\limits_{n=0}^{\infty} z^{3^n}$? Would the ...
2
votes
1answer
2k views

Uniform Convergence of $nx(1-x^2)^n$ on $[a,1]$

Hello I am trying to understand why the sequence of functions $f_n=nx(1-x^2)^n$ does not converge uniformly to 0 on the interval $[0,1]$ but does on the interval $[a,1]$ where $a\in (0,1)$. I know ...
7
votes
3answers
293 views

A property of some sequences of natural numbers (and their binary representation)

Let's consider a sequence of natural numbers $a_n$, represented in binary, with the following properties: $\forall n \in \mathbb{N}$ the number $a_n$ is represented with $n$ binary digits $\forall ...
3
votes
2answers
702 views

Equivalence of Completeness Axioms of Real Numbers

There are many equivalent versions of completeness in the real number system: i) LUB/supremum property ii) Monotone Convergence property iii) Nested Interval property iv) Bolzano Weierstrass property ...
2
votes
3answers
801 views

Cauchy Sequence of Real Numbers

Give an example of a sequence $(x_n)$ of real numbers, where $\displaystyle\lim_{n\to+\infty}|x_n-x_{n+1}|=0$, but $(x_n)$ is not a Cauchy sequence
6
votes
1answer
250 views

Finding the exact value of $ \sum \frac{4n-3}{n(n^2-4)} $

I would like to find the exact value of the series $$\begin{align*} \sum_{n=3}^{\infty} \frac{4n-3}{n(n^2-4)} \end{align*}$$ which is certainly a telescoping series. Do you have any idea of ...
21
votes
2answers
679 views

How to prove that $\sum\limits_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum\limits_{k=1}^\infty \frac{1}{(a+k)^2}$ for $a>-1$?

A problem on my (last week's) real analysis homework boiled down to proving that, for $a>-1$, $$\sum_{n=1}^\infty\frac{(n-1)!}{n\prod\limits_{i=1}^n(a+i)}=\sum_{k=1}^\infty \frac{1}{(a+k)^2}.$$ ...
1
vote
2answers
450 views

The first four terms x, y, z, w of an arithmetic sequence

I have been attempting this question for the past 3 days with no luck: ...
1
vote
3answers
507 views

Use the definition of a limit/triangle inequality to show divergence

I just asked a question about this kind of stuff so I feel bad asking again, but I could use some help. This is a homework question that reads: Use the definition of limit to prove that the sequence ...