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

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Limit ratio test

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) ...
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1answer
27 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
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1answer
28 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 ...
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1answer
48 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 ...
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4answers
45 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 ...
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1answer
50 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 $ ...
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2answers
82 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 = ...
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1answer
47 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 ...
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1answer
111 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 ...
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0answers
25 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 ...
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2answers
93 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 ...
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3answers
304 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*} ...
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1answer
48 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 ...
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2answers
324 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 ...
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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 ...
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1answer
19 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 ...
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1answer
49 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 ...
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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.
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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 ...
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1answer
19 views

Convergence of a Cauchy sequence of matrices

I have a Cauchy sequence of matrices $C_i \in R^{p \times q}$, i.e. $\lim_{n\rightarrow \infty} \| C_{n+1} - C_{n} \| = 0$ for any norm (I just need the property that $\|C_1-C_2\|>\delta ...
3
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2answers
53 views

Finding the convergence

The sequence $a_n = \sqrt[n]{(3^n+5^n)}$ is convergent? Tried to resolve applying the the limit $n\to\infty$ but couldn't figure out how to finish it, any idea? Thanks
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4answers
740 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 ...
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2answers
57 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 ...
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0answers
25 views

Integral/infinite sum related to Bessels which pop up in optical coherence theory

In propagating partially coherent optical fields, the following integral pops up: $I_1=\int_0^{2\pi} e^{i(a\cos[\theta]+b\cos^2[\theta])}d\theta$, where a and b are real numbers. If we consider ...
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1answer
24 views

$\ell^p$ spaces' inclusion

$$ \ell^s\subsetneq \bigcup_{k<p}\ell^k\subsetneq \ell^p\subsetneq\bigcap_{k>p}\ell^k\subseteq \ell^q $$ for any $1\le s<p<q$. Any idea to prove these inclusions? Counterexamples for the ...
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0answers
53 views

Does the sequence is convergent? [on hold]

Prove or disprove the convergence of sequence $(v_n )_{n=1}^{\infty}$ where $$v_n =n\sum_{k=1}^{\infty} \frac{1}{2^k }\left(1-\frac{1}{2^k}\right)^n .$$
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4answers
60 views

Why does $\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$ converge conditionally?

Why does it converge conditionally? $$\sum_{k=1}^{\infty} \frac {(-1)^{k-1}}{k}$$
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4answers
63 views

How to prove {$a_n$ } is increasing where $a_1 = \sqrt{2}$ and $a_{n+1} = \sqrt{ 2+ a_n}$ [duplicate]

I already found out that this sequence is bounded above and $a_n <2 \forall n \in \mathbb Z_+ $ I think I'm missing a point as I can't think of a way to prove that the sequence is increasing.
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2answers
48 views

If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too?

I got this question: Prove or disprove the following: If the series $\sum_{k=1}^{\infty} a_k3^k$ diverges, Must the series $\sum_{k=1}^{\infty} a_k4^k$ diverge too? I tried to find a couple of ...
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1answer
28 views

Radius of convergence of series given radius of convergence of another series

I'm hoping someone might be able to verify my solution to the following problem: Suppose that the series $\sum c_n z^n$ has radius of convergence $R$. Find the radius of convergence of the $\sum ...
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2answers
47 views

Proving that a convergent sequence has a unique limit

Is the following method wrong? Let {$a_n$} be a convergent sequence Assume $ \lim_{n \rightarrow \infty} \{a_n \}$ = L and $ \lim_{n \rightarrow \infty} \{a_n \}$ = M L-M =$ \lim_{n \rightarrow ...
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0answers
42 views

Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
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4answers
49 views

How to apply the alternating series test to the series $\sum (-1)^{n+1} n/2^n$?

So, I need to test the following series for convergence or divergence: $$\sum_{n=1}^\infty (-1)^{n+1}{n\over {2^n}}$$ I know that when you use the Alternating Series Test, the series must satisfy ...
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0answers
16 views

Recurrence relation-Summation of a series [on hold]

Sir, I have a Converging recurrence relation given as below, $-(\psi(n-1)) ...
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2answers
80 views

How to simplify this summation

I was wondering how to solve this infinite sum. $$\sum_{k=0}^\infty {1\over 4!} \cdot {k^7\over2^k}$$ I know roughly that for $$\sum_{k=0}^\infty {k \over 2^k}$$ the sum takes advantage of the ...
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1answer
43 views

Function whose power series coefficients contain logarithms

Is there a function that can be expressed as a power series $$f(x)=\sum_{n=0}^\infty a_n x^n$$ whose coefficients $a_n$ are expressions containing $\log n$ or something similar?
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0answers
23 views

convergence interval of an infinite series without the general term

I am trying to find the convergence of an infinite series of which I do not have the nth term. Instead of applying the ratio test for the nth term, I divided the first two terms, then the next two and ...
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1answer
93 views

Simplify this summation

$$\sum_{k=5}^\infty{{k-1}\choose{k-5}}\frac{k^3}{2^k}$$ I can't seem to simplify this sum. I get to a certain point then I get stuck, I know there must be some sort of trick to simplify it but I am ...
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0answers
64 views

Sum of series with two binomial coefficients [on hold]

How can I find sum of a series $$\sum_{k=m}^{\infty} {k \choose m} {k+n \choose n}x^k$$ where $x<1$ and $m,n$ are constants. Using wolfram mathematica it is obtained $$\sum_{k=m}^{\infty} {k ...
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0answers
27 views

Equality of a recurrent sequence and of a running maximum of another sequence

Let $\{a_n\}$ be a sequence of real numbers. Let $c,b$ be real constants. Define $$ L_{k,n}=\exp\left\{c\sum_{i=k}^n(a_i+b)\right\}. $$ Then it can be shown that $L_n=\max_{1\le k\le n}L_{k,n}$ is ...
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6answers
213 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 ...
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2answers
58 views

Prove or disprove $\sum (a_n + b_n) $ is divergent if $\sum a_n $ and $\sum b_n $ are divergent.

I proved it as follows. Since $\sum a_n$ and $ \sum b_n $ are divergent, $ \forall \epsilon > 0, \exists p \in \mathbb Z_+ st, n \gt p \implies \sum a_n > \epsilon \gt \frac{\epsilon}{2} $ ...
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1answer
177 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: ...
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45 views

Convergence/divergence of $\sum_{n=2}^{ \infty} [ (1+\frac{1}{\log n } )^{1/n}-1 ]$ [on hold]

Determine whether $$\sum_{n=2}^{ \infty} \left[ \left(1+\frac{1}{\log n } \right)^{\large\frac1n}-1 \right]$$is divergent or not. I tried the ratio test and the limit comparison, but they did not ...
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1answer
41 views

How to define divergence and prove $\sum r.a_n$ is divergent if $\sum a_n$ is divergent [on hold]

The first question was to prove or disprove $\sum ra_n$ is divergent if $\sum a_n $ is divergent (r $\in \mathbb R$ and r$\neq$0). I came across another problem when I was trying it, can I write the ...
6
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3answers
202 views

Solving recurrence relation: Product form

Please help in finding the solution of this recursion. $$f(n)=\frac{f(n-1) \cdot f(n-2)}{n},$$ where $ f(1)=1$ and $f(2)=2$.
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4answers
98 views

Why these two series are convergent or divergent?

I do not understand why $$\sum^{\infty}_{k=1} z_k = \sum^{\infty}_{k=1} \frac1k$$ is divergent but the other series $$\sum^{\infty}_{k=1} z_k = \sum^{\infty}_{k=1} \frac{(-1)^{k+1}}k$$ is convergent. ...
12
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1answer
103 views

A series involving inverses of harmonic numbers

How would I solve this question: If $$E_n = \frac{1}{2} + \frac{1}{4} + \frac{1}{6}+ \cdots +\frac{1}{2n}$$ and $$A_n = (2n+1)(E_n)(E_{n+1})$$ Find $$\sum_{n = 1}^{\infty}\frac{1}{A_n}$$ My try: ...
2
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2answers
113 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...
3
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0answers
69 views

How do you use the BBP Formula to calculate the nth digit of π?

I know what the Bailey-Borweim-Plouffe Formula (BBP Formula) is—it's $\pi = \sum_{k = 0}^{\infty}\left[ \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6} ...