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

learn more… | top users | synonyms (5)

0
votes
1answer
38 views

Prove that $T_n$ satisfy $ \sum_{k=0}^{N-1}{T_i(x_k)T_j(x_k)} = \begin{cases} 0 &: i\ne j \\ l\neq 0 &: i=j \end{cases} \,\! $

The Chebyshev polynomials of the first kind satisfy the recurrence relation $$ \begin{cases} T_{n}(x)=2xT_{n-1}(x)-T_{n-2}(x) \qquad n \geq 2 \\ T_{0}(x)=1, \ \ T_{1}(x)=x \\ \end{cases} $$ The ...
0
votes
2answers
69 views

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\sum_{n=1}^{\infty}a_n$ converges then $\sum_{n=1}^{\infty}\sin{(a_n)}$ converges.

Show that if $a_n > 0$ for all $n\in\mathbb{N}$ and $\displaystyle \sum_{n=1}^{\infty}a_n$ converges then $\displaystyle \sum_{n=1}^{\infty}\sin{(a_n)}$ converges. I conjecture that the final term ...
0
votes
1answer
64 views

Recurrence relation: $c_{k+1}=c_k+\frac{1}{(k+1)!}$

I have no idea how to proceed solving a recurrence relation like this. I know that the terms approach $e$ but beyond that I have no idea. The relation is$$c_{k+1}=c_k+\frac{1}{(k+1)!} \ \ ; \ c_0=1$$ ...
3
votes
3answers
134 views

If $ \sum_{n=1}^{\infty}x_na_n $ converges when $x_n\to 0,$ then $ \sum_{n=1}^{\infty}a_n $ also converges. [duplicate]

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence with non negative terms so that for every sequence $\{x_n\}_{n=1}^{\infty}$ with $x_n \geq 0$ and $\lim_nx_n=0$, the series $\sum_{n=1}^{\infty}x_na_n$ ...
4
votes
1answer
77 views

Baby Rudin exercise 3.3 solution, possible typo in solutions manual?

Okay so I'm working through the exercises in Rudin and after checking my solutions manual for 3.3, I found something that seems like it can't be true. Here is the original question in rudin: ...
7
votes
2answers
192 views

Show that the series converges, but not absolutely.

Show that the series converges, but not absolutely. $\sum_{n=1}^{\infty}( $exp$(\frac{(-1)^n}{n})-1)$. My Try: Let $a_n=$exp$(\frac{(-1)^n}{n})-1$. I was going to use alternating series test ...
3
votes
2answers
87 views

How to replace addition with multiplication to find the next integer value?

Sorry in advance for my lack of mathematical knowledge, I am very new to it. Yesterday, I posed this question to myself: "In a world without addition or subtraction, how could we derive the next ...
1
vote
1answer
24 views

For which complex values of $s$ does $\left\{n^{−s}\right\}$ belong to $l^2$? [closed]

Would I be correct in saying that for all s in R the values s<=0 hold? and if so does it hold for all s in C? For which values of $s \in \mathbb{C}$ does $\left\{n^{−s}\right\}$ belong to $l^2$? ...
1
vote
0answers
32 views

Special Non-linear recurrence

Problem I have a non-linear recurrence relation given by $$ a_n = a_{n-1}+a_{n-2}+a_{n-3} - \sqrt{a_{n-1}.a_{n-2}+a_{n-2}.a_{n-3}+a_{n-3}.a_{n-1}} $$ Given $ a_1, a_2 $ and $ a_3 $,I have to find ...
2
votes
1answer
43 views

$f_n(x)$ converges uniformly to a function $f(x)$ then does it follow that the limit function $f(x)$ is also uniformly continuous.

If a sequence of functions $f_n(x)$ converges uniformly to a function $f(x)$, and if each $f_n(x)$ is uniformly continuous, then does it follow that the limit function $f(x)$ is also uniformly ...
2
votes
1answer
28 views

$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.

Assume that $f_n → f$ uniformly on $S$ and each $f_n$ is continuous on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$. I'm stuck in thinking about it ...
4
votes
1answer
136 views

Closed form for this sum?

$$\displaystyle \sum_{m=0}^{\infty}{\frac{{\left({H}_{m}^{(1)}\right)}^{2} - {H}_{m}^{(2)}}{{(m+1)}^{6}}}$$ where $ \displaystyle {{H}_{k}}^{(r)} = \sum_{i=1}^{k}{\frac{1}{{i}^{r}}} $ I have no ...
1
vote
0answers
31 views

Sum of infinite dimensional random variables

If $X_i$ are Independent and Identically Distributed (IID) vector valued variables with positive mean, and finite variance, then with Chebyshev's inequality, we know that their sum $S_n=\sum_{k\leq n} ...
0
votes
1answer
33 views

Trouble with series question from STEP past paper

I have answered all parts of this question but the last part. By using the identity, $\cot x - \tan x = 2\cot 2x$ ...
2
votes
1answer
58 views

Find product limit of this recursively-defined sequence?

Problem: if $a_1=3$, $a_n=2a_{n-1}^2-1$, $n\ge2$, find the limit of this expression: $$\lim\limits_{n \to ∞} \prod\limits_{k=1}^{n-1} (1+\frac {1}{a_k})$$ The original problem asks to find ...
0
votes
1answer
30 views

How can I solve the following exercise

Prove that a linear operator $T:X\rightarrow Y$ is bounded if and only if it maps sequences that converge to zero to bounded sequences .
0
votes
1answer
22 views

Problem in finding the first term and common difference

The sum of the first $6$ terms of an arithmetic sequence is $96$. The sum of the first $10$ terms is one third of the sum of first $20$ terms. I tried to equate the sum of the first $10$ terms and ...
0
votes
5answers
34 views

Finding the first 3 terms

The sum of the first $3$ terms of a geometric sequence is $65$ and their product is $3375$. How do I find the first three terms? I know that the answer is $5$, $15$ and $45$, but I don't know how to ...
0
votes
1answer
40 views

Convergent conjecture: What is the proof?

Lets say that $\def\nn{\mathbb{N}}$$\def\rr{\mathbb{R}}$$K : \nn \to \rr$ and $\displaystyle \sum_{i=1}^\infty \frac{K(i)}{K(i+1)}$ is a convergent sum. My conjecture is that the function $K$ must be ...
4
votes
2answers
205 views

Finding a recurrence for a sum

I am trying to implement the following sum using a programming language: $$\sum_{i=1}^N a^i i^r$$ where $N$, $a$ and $r$ are integers. The problem is, I cannot find a suitable way to do this. ...
1
vote
0answers
26 views

the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$.

To show that the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$. My Try: Let us consider $u_n(x) =\cos(nx), ...
3
votes
2answers
74 views

Investigate the convergence of $\sum a_n$, where $a_n =\int_{0}^{1} \frac{x^n \sin(\pi x)}{1-x} dx$.

Problem: Investigate the convergence of $\sum a_n$, where $a_n =\int_{0}^{1} \frac{x^n \sin(\pi x)}{1-x} \, \mathrm{d}x$. I'm thinking about changing $\frac{1}{1-x}$ to $\sum x^k$ and then ...
2
votes
3answers
336 views

Why does this sum converge?

I know that the following sum converges to 2 via WolframAlpha, but I am not sure why. $$\sum_{k=1}^\infty k \left[\frac{2}{k} - \frac{4}{k+1} + \frac{2}{k+2}\right] = 2$$ WolframAlpha gives the ...
3
votes
2answers
88 views

How do you determine if a number is a even Fibonacci number or not? [duplicate]

Rather than computing out the whole Fibonacci sequence and check if $n$ is even and in there, is there a more straightforward way to compute if $n$ is a even Fibonacci number?
1
vote
2answers
43 views

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n

Given that $\tan x=\sum_{i=0}^{\infty}a_nx^n$, Show that $a_n=0$, for even n. from the series expansions of $\sin x$ and $\cos x$, I get that $\tan ...
2
votes
0answers
55 views

Compute $\sum_{b=2}^{\infty}{\left[\sum_{k=1}^{\infty}{\left(\frac{digitsum_b(k)}{k(k+1)}+\left(1+\frac1b\right)\frac{(-1)^k}{kb^k}\right)}\right]}$

Warning: This post contains more than one question and is pretty long. I decided include them all in this post, because they all emerged from the same question. Furthermore, I decided to display all ...
5
votes
1answer
112 views

A limit of a sequence statistifying $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})=a_1+a_2+…+a_n$

Sequence $\{a_n\}$ is a positive sequence and satisfies $S_{n} = \frac{1}{2}(a_{n}+\frac{1}{a_{n}})$ where $S_n = a_1+a_2+...+a_n$. Find $\lim_{n\to \infty} S_{n+1}*(S_{n}-S_{n-1})$
3
votes
2answers
70 views

$\int_a^b f(x)g(x)dx = \sum \int_a^b f_n(x)g(x)dx.$

Let $\sum f_n(x) $ be uniformly convergent to $f(x)$ on $[a,b]$ where each $f_n$ is continuous on $[a,b]$. If $g: [a,b] \to \mathbb R$ be integrable on $[a,b]$, then $$\int_a^b f(x)g(x)dx = \sum ...
0
votes
0answers
51 views

If $\sum a_n$ is a convergent series of real numbers then the following series are convergent.

If $\sum a_n$ is a convergent series of real numbers prove that the series: 1) $\sum a_n e^{-nx}$ is uniformly convergent on $[0,\infty)$; 2) $\sum \frac{a_n}{n^x}$ is uniformly convergent on ...
1
vote
2answers
157 views

Proof of the reciprocal of all semiprimes diverging?

$$\sum_{\text{semi-primes}}\frac{1}{s}=\frac{1}{4}+\frac{1}{6}+\frac{1}{9}+\frac{1}{10}\cdots$$ I almost positive that this sum diverges, but I would really like to see a very thorough proof. Thank ...
4
votes
2answers
147 views

Evaluating $\sum_{n=0}^{\infty } 2^{-n} \tanh (2^{-n})$

Reading in some tables pages I found $$\sum _{n=0}^{\infty } 2^{-n} \tanh \left(2^{-n}\right)=\tanh (1) \left(1+\coth ^2(1)-\coth (1)\right)$$ I try to split in two sum using the roots of the ...
1
vote
1answer
85 views

What does this infinite product come out to?

$$1\cdot \frac{1}{2}\cdot 3\cdot \frac{1}{4}\cdot 5\cdot \frac{1}{6}\cdots$$ What does this product come out to? It does diverge, but products like this tend to have values $\lt \infty$. Here is what ...
1
vote
2answers
53 views

Maclaurin series for $f(x) =\pi x^8 e^{-x^3}$

Find the Maclaurin series for: $$f(x) =\pi x^8 e^{-x^3}$$ What I have : $e^x = $ $\sum_{n=0}^\infty {x^n\over n!}$ THEREFORE $x^8 e^{-x^3}= $ $$\sum_{n=0}^\infty {-x^{3n+8}\over n!}$$ Now I don't know ...
1
vote
0answers
48 views

Equating two definitions of Zeta function

The Zeta function $\zeta(s)$ is defined as following $$ \zeta(s)=1+\frac1{2^s}+\frac1{3^s}+\frac1{4^s}+\frac1{5^s}+\cdots=\sum_{n=1}\frac1{n^s} $$ Now it has been shown that $$ \tag1 ...
3
votes
3answers
90 views

What is the value of $\sum e^{-n} \sin^2 n$?

Clearly the series $\sum_1^\infty e^{-n} \sin^2 n$ converges. If I put it into Maple, I get an exact value: $$ -\frac {{\rm e}^1 ( {\rm e}^1 (\cos(1))^2 + (\cos(1))^2 - {\rm e}^1 -1 ) }{-4{\rm e}^2 ( ...
1
vote
2answers
52 views

Faulhaber Formula Identity

I have to show the following identity: $$ S_n^p := 1^p+2^p+...+n^p $$ $$ (p+1)S_n^p+\binom{p+1}{2}S_n^{p-1}+\binom{p+1}{3}S_n^{p-2}+...+S_n^0=(n+1)^{p+1}-1 $$ What I did first is to use the binomial ...
1
vote
0answers
19 views

Covering a family of sets of $\mathbb{Z}^d$ with boxes of a given diameter

Let $diam(A)$ be the graph distance between the two farthest vertices contained in a finite set $A \subset \mathbb{Z}^d$. Is it true that there exists a real $\eta(d)>0$ such that for any finite ...
1
vote
1answer
45 views

Breaks the product based on ${x_j}$?

Can anyone help me to break this product into the series based on ${x_j}$ ? $$ \prod_{i=1}^{K}(1-x_i)$$ I want to break it to some function as below: $$ \sum_j\Psi(x_j) $$ I saw something like ...
1
vote
1answer
40 views

Limit of continuous convex functions.

Let $\left( \, f_n(x)\, \right)$, $x \in \mathbb{R}$, be convex continuous increasing functions. Let $$f(x) := \lim\limits_{n \rightarrow \infty} f_n(x)$$ and assume that the limit exists for every ...
1
vote
2answers
21 views

Monotonicity of this series

Given the function $f:\mathbb{R}\to \mathbb{R}$ with $f(x)=e^{x-1}$ we have the series $(x_n)_{n\ge1}$ where: $x_1=2$ and $x_{n+1}=f(x_n), n\ge1$ Find the monotonicity of the series and compute ...
2
votes
2answers
97 views

Finding the Sum of a series $\frac{1}{1!} + \frac{1+2}{2!} +\frac{1+2+3}{3!}+…$

I need to find the sum of this series $\dfrac{1}{1!} + \dfrac{1+2}{2!} + \dfrac{1+2+3}{3!}+...$ But somehow I am not even convinced this converges. I tried writing it as $\sum \dfrac{n(n+1)}{2(n!)}$. ...
1
vote
1answer
66 views

How to evaluate this limes?

$a_0 = 0$ $a_n = \sqrt{2+a_{n-1}}$ $$\lim\limits_{n\rightarrow \infty} 2^n \sqrt{2-a_{n-1}} =\ ?$$ I don't even know where to start.
5
votes
1answer
84 views

How find $\lim_{n\to+\infty}\sum_{k=0}^{n}(-1)^{k}\sqrt{\binom{n}{k}}$?

How find this limit $\displaystyle\lim_{n\to+\infty}\sum_{k=0}^{n}(-1)^{k}\sqrt{\binom{n}{k}}$
6
votes
3answers
100 views

Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$

Given the Fibonacci, tribonacci, and tetranacci numbers, $$F_n = 0,1,1,2,3,5,8\dots$$ $$T_n = 0, 1, 1, 2, 4, 7, 13, 24,\dots$$ $$U_n = 0, 1, 1, 2, 4, 8, 15, 29, \dots$$ and so on, how do we show ...
0
votes
2answers
48 views

Summation of a series of Positive Prime numbers

Is there a way to find the sum for a set of positive prime numbers (e.g., the first $25$ prime numbers) without just arbitrarily adding them up shorthand?
2
votes
5answers
127 views

How to show that $\int\limits_1^{\infty} \frac{1}{x}dx$ diverges(without using the harmonic series)?

I was reading up on the harmonic series, $H=\sum\limits_{n=1}^\infty \frac{1}{n}$, on Wikipedia, and it's divergent, as can be shown by a comparison test using the fact that ...
1
vote
1answer
38 views

Formula for giving always the same sequence of numbers

I have a variable in excel that starts at 1 and it keeps incrementing indefinitely (i named it lr). I want that when lr gets to 21 it stop getting it's actual value and restart valuing 1. When it ...
0
votes
2answers
61 views

A Series might be a number or a sequence – Is there a better notation?!

Take the expression $\sum_{k=1}^\infty a_k$. Sometimes this expressions refers to the sequence of partial sums $\left(\sum_{k=1}^n a_k\right)_{n\in\mathbb N}$ and sometimes to the limit of this ...
4
votes
1answer
47 views

Recurrence relation in terms of another sequence

How do I solve a recurrence of the form $$nd^{n-1}a_n+a_{n+1}d^{n+1}=b_n$$ for $a_n$, where $b_n$ is another (known) sequence and $d$ is a constant? My only idea was to use a generating function and ...
2
votes
1answer
74 views

Find the sum of the infinite series [1+(5/1!)+(8/2!)+(11/3!)+…]

Excluding the 1st term of the series 1, if we start from the 2nd term-($\frac{5}{1!}$), I can locate that the numerators are in A.P with common difference 3, & 1st term 5. Whereas the ...