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

learn more… | top users | synonyms (3)

4
votes
1answer
38 views

Speeding up the convergence of a series

I want to speed up the convergence of a sequence involving rational expressions the expression is $$\sum _{x=1}^{\infty }\left( -1\right) ^{x}\dfrac {-x^{2}-2x+1} {x^{4}+2x^{2}+1}$$ If I have not ...
1
vote
1answer
22 views

A problem on AP and GP [on hold]

let $a,x,b$ as well as $c,x,d$ are in GP while $a^{2},y,b^{2}$ as well as $c^{2},y,d^{2}$ are in AP. How can I prove that $a^{n} + b^{n} = c^{n} + d^{n}$ when $n$ is an even integer?
4
votes
3answers
44 views

Cannot recongize sequence

D12/22. What is the missing number? $$\frac{1}{16}, \frac{1}{8}, \frac{3}{16}, \frac{1}{4}, \frac{5}{16}, \ \ \ [?]$$ $$A. \frac{5}{4}$$ $$B. \frac{3}{4}$$ $$C. \frac{5}{8}$$ $$D. \frac{3}{8}$$ ...
0
votes
2answers
43 views

Finding a closed form for $x_{k+1}=\frac{1}{3}+\frac{k-1}{2}+x_k$ [on hold]

I have a sequence $\{x_n\}$ such that $x_2=\frac{7}{6}, x_3=\frac{5}{2}$ and $x_{k+1}=\frac{1}{3}+\frac{k-1}{2}+x_k$. I want to find the $x_l$. I know that this is a problem of recurrence relation. ...
0
votes
2answers
45 views

Find a sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$

Find a sum of $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{ch(n)}{3^n}$$ Could you give some some hint or some way to start this? I have tried representing ch(n) through its definition with e, but I ...
2
votes
1answer
40 views

Proving there exist convergent subsequences for bounded sequence of real numbers

I'm trying to teach myself some basic topology by self-studying from Intro to topology by Mendelson. I'm stuck on one of the exercises and can't figure out how to proceed with the proof. The question ...
2
votes
3answers
54 views

Does $\sum_{n=1}^{\infty}\frac{n-1}{n^2}$ converge or diverge?

Is my logic OK? $a_{n}=\frac{n-1}{n^2}$ $\frac{1}{n} \leq b_{n}=\frac{n-\frac{n}{2}}{n^2}=\frac{n}{2n^2}=\frac{1}{2n} \leq a_{n}=\frac{n-1}{n^2}$ and there for the initial series diverges.
2
votes
2answers
52 views

Faulhaber-like series

The series $$F(n,m) := 1^n + 2^n + \ldots + m^n = \sum_{k=1}^m k^n$$ Is known as the Faulhaber's Series. I've tried to find a formula for this similar series but I've failed so far. ...
57
votes
5answers
9k views

Help me solve my father's riddle and get my book back

My father is a mathteacher and as such he regards asking tricky questions and playing mathematical pranks on me once in a while as part of his parental duty. So today before leaving home he sneaked ...
3
votes
2answers
44 views

Prove that $\exists \{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ coverges.

Given a real positive sequence $\{a_n\}$ such that $\sum_{i=1}^n a_i$ converges. Prove that there exists a real sequence $\{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty ...
0
votes
1answer
18 views

Chong inequalites about permutations

I read about two inequalities called Chong's inequalities. They state: $$\sum_{k=1}^N\dfrac{a_k}{a_{\pi(k)}}\ge N$$ and $$\displaystyle\prod_{k=1}^Na_k^{a_k}\ge\prod_{k=1}^N a_k^{a_{\pi(k)}}$$ I ...
0
votes
1answer
64 views

distance travelled after nth bounce

A ball is thrown vertically to a height of $625$ meters from ground. Each time it hits the ground it bounces $\frac{2}{5}$ of the height it fell in the previous stage. How much will the ball travel ...
0
votes
2answers
48 views

When $\lim\left| a_{k+1}/a_k \right| =1$ in the limit ratio test, does it follow that the series diverges?

In my lecture notes the theorem for the ratio test goes as follows: Let $(a_k)_{k\in \mathbb{N}}$ be a sequence of real numbers with $a_k \neq 0, \forall k \geq \hat{k} \in \mathbb{N}$. (a) ...
1
vote
1answer
33 views

Infinite series for recurrence

Question 1 If I define $A(z) = \sum_{n \ge 0} a_n \frac{z^n}{n!} \tag 1$ (where $a_n$ are $3\times 3$ constant matrices indexed with n), then can we re-write $\sum_{n \ge 1} a_{n-1} \frac{z^n}{n!} ...
3
votes
1answer
32 views

Prove that $(x_n)_{n\geq1}$ is an arithmetic progression

Let $(x_n)_{n\geq1}$ be a sequence of integers. Define $y_n=\frac{x_n}{n},n\geq1$. The sequence $(y_n)_{n\geq1}$ is convergent and $n$ divides the sum of any $n$ consecutive terms of the sequence ...
2
votes
6answers
217 views

Algebraic proof of $\tan x>x$

I'm looking for a non-calculus proof of the statement that $\tan x>x$ on $(0,\pi/2)$, meaning "not using derivatives or integrals." (The calculus proof: if $f(x)=\tan x-x$ then $f'(x)=\sec^2 ...
0
votes
1answer
186 views
+100

Logic of numerical series

One of our colleagues has written a numerical series to the whiteboard in our breakroom. Nobody until now could solve the logic behind this series: ...
10
votes
3answers
306 views

Series for logarithms

This is more of a challenge than a question, but I thought I'd share anyway. Prove the following identities, and prove that the pattern continues. \begin{equation*} ...
6
votes
2answers
129 views

Convergence of $\sum_{k=1}^{\infty} \left (\sum_{j=1}^{k}\frac 1 j\right)^{-k}$

Does $\displaystyle\sum_{k=1}^{\infty} \left (\sum_{j=1}^{k}\frac 1 j\right)^{-k}$ converges ? Let's call the inner sum $a_k$ such that $\displaystyle\sum_{k=1}^{\infty} (a_k)^{-k}$, applying ...
1
vote
1answer
21 views

Testing the convergence of the series $\sum 1/(k^q (\ln k)^p)$

Determine all values of $p$ and $q$ for which the following series converges: $$\sum_{k=2}^{\infty} \frac{1}{k^q (\ln k)^p}$$ Hints : Consider the three case $q>1$, $q=1$, $q<1$. I ...
1
vote
1answer
49 views

Does this convergence test for series hold?

Long ago before I joined math.se, a friend asked this question here after he and I had discussed it some. An answer was accepted that is narrow in scope, so I am going to ask a more specific version ...
2
votes
0answers
134 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
0
votes
3answers
48 views

Lower bound of an inductively defined sequence

We have a sequence $U_{n}$, defined as : $U_0=5$, and $U_{n+1}=\frac{1}{2}(U_n+\frac{5}{U_n})$. I'm trying to prove that for each $n \in \mathbb{N}$, $U_n \geq \sqrt{5}$. I've tried to prove that ...
0
votes
2answers
85 views

The series $\sum a_n$ converges, where $a_n$ is the product of fractions from $1/2$ to $(2n-3)/(2n-2)$, divided by $2n-1$

Prove the series given by the sequence $$a_n= \frac{1}{2}·\frac{3}{4}·\ldots ·\frac{2n-3}{2n-2}·\frac{1}{2n-1}$$ converges The series is $$\sum_{n=1}^\infty a_n = ...
0
votes
4answers
46 views

Finding general formula for a recursion function

If we have a recursion relation defined as $a_n = 3a_{n-1}+1$ with $a_1=1$ then find the general formula for $a_n$ in terms of $n$ with a(1) = 1. So far I have: $a_n = 3a_{n-2}+1+1 = 3a_{n-3}+1+1+1 ...
2
votes
3answers
78 views

The sum of geometric series $e^{k-1}/\pi^{k+1}$

Let $T_n=\sum _{k=1}^{n}\dfrac{e^{k-1}}{\pi ^{k+1}}$ calculate the $\lim_{n\to\infty}T_n$ Note $T_n$ is a geometric series: \begin{align*} T_n&=\sum _{k=1}^{n \:}\dfrac{e^{k-1}}{\pi ...
2
votes
2answers
64 views

Polygamma function series: $\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2$

Applying the Copson's inequality, I found: $$S=\displaystyle\sum_{k=1}^{\infty }\left(\Psi^{(1)}(k)\right)^2\lt\dfrac{2}{3}\pi^2$$ where $\Psi^{(1)}(k)$ is the polygamma function. Is it know any ...
0
votes
1answer
51 views

Prove that {$b_n$} is convergent with $ b_n \to L$ [duplicate]

Let $\lbrace a_n\rbrace$ be a convergent sequence with $a_n \to L$ Define $$ b_n = \frac{ a_1 + a_2 + ... a_n}{n} \forall n \in \mathbb Z_+ $$ Prove that $\lbrace b_n\rbrace$ is convergent with $ ...
13
votes
6answers
2k views

A strange puzzle having two possible solutions

A friend of mine asked me the following question: Suppose you have a basket in which there is a coin. The coin is marked with the number one. At noon less one minute, someone takes the coin ...
7
votes
5answers
404 views

Definite integral, quotient of logarithm and polynomial

I was thinking this integral : $$I(\lambda)=\int_0^{\infty}\frac{\ln ^2x}{x^2+\lambda x+\lambda ^2}\text{d}x$$ What I do is use a Reciprocal subsitution, easy to show that: ...
0
votes
1answer
114 views

Why is (European) money in units of $1,2,5,10,20,50, \cdots\;$? [on hold]

In the old days, in the Netherlands, we had 1 ct (cent), 5 ct (stuiver), 10 ct (dubbeltje), 25 ct (kwartje), 1 gld (gulden), 2.5 gld (rijksdaalder), 10 gld (tientje), ... And then they decided we ...
0
votes
1answer
48 views

common terms problem

Does anyone have an idea in finding common terms of two following sequences? \begin{matrix} x_0=2,x_1=12, x_{n+1}=6x_n-x_{n-1} \\ x'_0=8,x'_1=144,x'_{n+1}=18x'_n-x'_{n-1}\end{matrix} What is the most ...
1
vote
0answers
26 views

Prove something is a signed measure

Given a measure space $(X,\mathcal{M},\mu)$ and a measurable function $f:X\rightarrow \overline{\mathbb{R}}$ such that at least one of $f^+$ or $f^-$ is integrable, show that ...
2
votes
2answers
94 views

Why does $\sum_{n=1}^\infty \sqrt{n+1}-\sqrt{n}$ diverge?

Why does $\sum_{n=1}^\infty \sqrt{n+1}-\sqrt{n}$ diverge? Using the ratio test I get the following. First of all since ...
5
votes
2answers
328 views

A certain “harmonic” sum

Is there a simple, elementary proof of the fact that: $$\sum_{n=0}^\infty\left(\frac{1}{6n+1}+\frac{-1}{6n+2}+\frac{-2}{6n+3}+\frac{-1}{6n+4}+\frac{1}{6n+5}+\frac{2}{6n+6}\right)=0$$ I have thought of ...
1
vote
1answer
49 views

What is the value of a limit of sums with $\sin k$

What is the value of the following limit? $$\lim_{t\to \infty} \sum_{k=1}^{\lfloor 10^t π \rfloor} \sin k $$ I don't know what to do. I need your help. Thank you. P.S. I think series diverges ...
-1
votes
1answer
136 views

Solve a differential equation using the power series method

Problem By assuming a power series solution of the form $$y(x) = \sum_{m=0}^{\infty} c_mx^m , \quad c_0 \not =0 $$ Show that the equation $ 2y'+xy=x $ has general solution ...
10
votes
1answer
269 views

Do runs of every length occur in this string?

In reference to the strings defined here (constructed by repeatedly appending the last "half" of the current string), consider the particular infinite string $s$ generated by starting with ...
19
votes
1answer
269 views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

I discovered the following conjecture by evaluating the integral numerically and then using some inverse symbolic calculation methods to find a possible closed form: $$\int_0^\infty\frac{\ln ...
1
vote
1answer
20 views

Computing the radius of convergence of a given series

Could anyone help me with the following problem? I'm getting $1$ as the answer, but I found the solution (without justification) to this problem online and it says the answer is $1/2$. Determine ...
0
votes
1answer
50 views

Showing Taylor Series for $f(x) = e^{-x^2}$ converges to $f$

Show Taylor Series for $f(x) = e^{-x^2}$ converges to $f$ I am stuck because when taking the (n+1) th derivative of f, I do not see a general pattern. Meaning I am having difficulty in bounding ...
4
votes
4answers
746 views

Why don't we indicate the variable to summed as we do for integrals?

When integrating over a certain variable $x$, we make sure to end the integral with $dx$, like so: $$\int_{1}^{\infty}\frac{1}{x^2}dx$$ The reason for this of course becomes more clear as one goes ...
1
vote
0answers
25 views

How to research a series, when only some elements are available

I want something like the On-Line Encyclopedia of Integer Sequences, but for series, not sequences. I'd like to know the name of a series, in what natural phenomenon it happens and so on.
3
votes
1answer
26 views

Why does $\lim\limits_{N \rightarrow \infty}{\sum_{i=1}^{N}\frac{1}{\frac{N}{1-\epsilon}-i}}$ converge to $\log\left[\frac{1}{\epsilon}\right]$?

while playing around with my equations, i found that the following has to hold for my universe to be consistent: $$\lim_{N \rightarrow ...
20
votes
12answers
3k views

$ \lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $ \displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
17
votes
8answers
2k views

How I can prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2?

Prove that the sequence $\sqrt{2} , \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}$ converges to 2 My attempt I proved that the sequence is increasing and bounded by 2, can anyone help me show that the ...
19
votes
4answers
586 views

Limit of series involving ratio of two factorials

$$ \sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3} $$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
52
votes
13answers
5k views

How can I evaluate $\sum_{n=0}^\infty (n+1)x^n$

How can I evaluate $$ \sum_{n=1}^\infty \frac{2n}{3^{n+1}} $$ I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is ...
3
votes
1answer
92 views
+300

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
0
votes
3answers
160 views

Nth Term of Fractions

How do I work out the Nth term of these fractions? $$1,\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots$$ Would I need to change them all into decimals but that would be quite complicated and ...