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

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1
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4answers
430 views

Limit of a Recursive Sequence

I'm having a really hard time finding the limit of a recursive sequence - $$ \begin{align*} &a(1)=2,\\ &a(2)=5,\\ &a(n+2)=\frac12 \cdot \big(a(n)+a(n+1)\big). \end{align*}$$ I proved ...
1
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2answers
330 views

Find the 12th term and the sum of the first 12 terms of a geometric sequence.

A geometric series has a first term $\sqrt{2}$ and a second term $\sqrt{6}$ . Find the 12th term and the sum of the first 12 terms. I can get to the answers as irrational numbers using a calculator ...
1
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1answer
670 views

How to prove that the sequence $n^{1/n}$ converges to 1 [duplicate]

Help me in proving that the sequence $n^{1/n}$ converges to $1$
2
votes
1answer
77 views

Does the series $\sum_{k=1}^\infty b^{k^2}$ have a closed form?

Does the following series have a closed form solution ? $$S(b) = \sum_{k=1}^\infty b^{k^2}$$ Please point to resources, if any, that I could use to learn how to determine the above for series with ...
5
votes
2answers
1k views

Proving the AM:GM inequality

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
6
votes
4answers
560 views

proof for the Ramanujan's formula ?

I found this formula in a textbook in which the proof to the formula was not given Ramanujam's formula $$\sqrt{1 +n\sqrt{1 +(n+1)\sqrt{1 + (n+2)\sqrt{1 + (n+3)\sqrt{1 +....\infty}}}}} = n+1$$ Its a ...
0
votes
2answers
144 views

proof of convergence of a sequence

I cant figure out how to prove that this series is convergent. I have to prove that if a sequence $(a_n)$ is positive, and that the series $\sum_{n=1}^\infty a_n$ converges, then $\sum_{n=1}^\infty ...
1
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1answer
75 views

Solution of $\sum_{n=1}^{\infty}x^{a^n}$

I was doing some math problems when I got stuck at summing the series of the form $$\sum_{n=1}^{\infty}x^{a^n}$$ where $x<1$ and $a$ is any positive integer. Since $x^n>x^{a^n}$, the series ...
4
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5answers
267 views

How to calculate this expression?

evaluate the expression [1]: $$\sum\limits_{n = 1}^\infty {\left( {\frac{1}{n} - \frac{1}{{n + x}}} \right)} $$ where $x$ is a real number, $0\le x\le1$, and $x$ is rounded to 3 digits. For ...
0
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3answers
65 views

A seemingly easy sequence

Can anyone tell me about the sum of the series $\Sigma \frac{1}{(2n)(2n+1)}$.This is not a usual telescoping sum in which all the terms cancel out. The limit of sum is from n=1 to infinity.
1
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1answer
144 views

I need help finding a generating function using some relation between the Bessel function and the Laplace integral for the Legendre Polynomials

The Bessel function of the first kind and order n has the integral representation $J_n(z)=i^{-n}/\pi \int_0^\pi e^{iz\cos\theta}\cos(n\theta)d\theta$ By using the Laplace integral for the Legendre ...
0
votes
1answer
53 views

Clarification of proof on the completion of a metric space using Cauchy sequences

This is in reference to the proof of the completion theorem of metric spaces. (To protect against link rot, here is a copy of the document being referenced: page 1, page 2, page 3) A proof is ...
9
votes
3answers
238 views

Properties of $\bigcap_{p > 1} \ell_p$

Consider the following space of sequences $$\left\{a=(a_n)_{n\in\mathbb{N}}:a\in\bigcap_{p>1}\ell_p, a_n\in\mathbb{R}\right\}$$ What are some of its properties? What is its relation to $\ell_1$ and ...
5
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2answers
220 views

Is there any value of zeta that is an integer?

Is there any value which we can substitute for $s$ in $\zeta (s)$ such that $$\sum_{n=1}^{\infty }n^{-s}\in \mathbb{Z}$$
12
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2answers
438 views

prove $\sum\limits_{n\geq 1} (-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \zeta^2(2)/2-\frac{7}{4}\zeta(3)\log(2)$

Prove the following $$\sum\limits_{n\geq 1}(-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \frac{1}{2}\zeta(2)^2-\frac{7}{4}\zeta(3)\log(2)$$ I was able to prove the formula above and interested in ...
1
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2answers
298 views

Find an expression for the common difference in terms of $x$, $y$, and $n$

The sum of the 1st and 2nd terms of an arithmetic series is $x$ and the sum of the $(n-1)$th and $n$th terms is $y$. Show that the sum of the first $n$ terms is $(n/4)(x+y)$. Find an expression for ...
9
votes
3answers
219 views

Evaluate the series $\sum_{k\geq 1} \frac{1}{2^k k^2}$

How to prove that $$\sum_{k\geq 1} \frac{1}{2^k k^2}=\frac{\pi^2}{12}-\frac{1}{2}\log(2)^2$$ without using the well-known $\operatorname{Li}_2\left( \frac{1}{2} \right)$ ? Edited : Thanks for L.F ...
0
votes
1answer
94 views

Proving divergence with the LCT

How do I do this? According to my book there are only three cases; L > 0 means converges if one of the series converges L = $\infty$ converges if one of the series converges L = 0 if one of the ...
25
votes
1answer
475 views

Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let ...
0
votes
0answers
80 views

Function from series summation?

There are two variables in time series $Z$ and $S$ depending on $t$. They are formulated as summation of series: $$\sum_{t=1}^TZ_t=\sum_{t=1}^TS_t-\beta(S_1-S_2-S_{T-1}+S_T)$$ Is there a way to ...
5
votes
2answers
1k views

why can we interchange summations

Suppose we have the following $$ \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_{ij}$$ where all the $a_{ij}$ are non-negative. We know that we can interchange the order of summations here. My ...
3
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2answers
412 views

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

The above is the telescoping series and the terms cancel but I'm left with $-\arctan(1)$ which is equal to $-\pi \over 4$ which is not correct.
2
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0answers
386 views

Is this proof that every convergent sequence is bounded correct?

I've tried the following proof that a convergent sequence is bounded but I'm not sure if it is correct or not. Let $(M,d)$ be a metric space and suppose $(x_k)$ is a sequence of points of $M$ that ...
1
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1answer
75 views

If $a_n =\int^{\pi}_0 \frac{\sin(2n-1)x}{\sin x}dx$ ,

If $$a_n =\int^{\pi}_0 \frac{\sin(2n-1)x}{\sin x}dx$$ , then $$a_1,a_2,.....a_n$$ are in (a) A.P and H.P (b) A.P and G.P but not in H.P (c) G.P and H.P (d) A.P. ,G.P and H.P. I have ...
9
votes
0answers
244 views

To how many decimals is $\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$ correct?

Consider: $$\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$$ This is, as far as I'm able to check with my software, correct to at least 167 decimals. If anyone has the ...
1
vote
1answer
86 views

Sum containing primes

Can anybody compute the value of $$\sum_p\sum_{k=2}^\infty\frac{\log(p^k)}{k}-\sum_p\sum_{k=2}^\infty\frac{\sum\limits_{p^n<k}\log(p^n)}{k(k+1)}$$ I have tried a lot but cannot think about the ...
3
votes
4answers
263 views

How to determine if $\frac{2n-1}{3n+1}$ is bounded

I have to determine whether the above sequence is bounded (from above or below). Bounded from above means: $\frac{2n-1}{3n+1} \le M$ $\forall n$ Bounded from below means: $\frac{2n-1}{3n+1} \ge m$ ...
13
votes
4answers
378 views

Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$

In this thread a friend posted the following integral $$I=\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$$ The best we could do is expressing it in terms of Euler sums ...
3
votes
3answers
269 views

Find the sum of the geometric series: $\;\sum_{k = 4}^\infty \frac 2{3^k}$

Find the sum of the geometric series: $\quad \displaystyle \sum_{k = 4}^\infty \frac 2{3^k}$ I converted $\dfrac{2}{(3^k)}$ into $2(3^{-k})$ so since $|r|>1$ the series diverges so I can't find a ...
1
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0answers
131 views

Generalizations of Cauchy's root test

There are some convergence tests that involve applying certain functions or transformations on the generic term of a series in order to determine if it converges, and then taking a limit. $$\lim_{k → ...
0
votes
1answer
67 views

How to compute the formula of $S_n$

$S_1$=a, $S_2$=b, $S_n$=|$S_{n-1}$-$S_{n-2}$|(n $\ge$3). Can I compute the formula of $S_n$? Thanks in advance.
4
votes
1answer
259 views

Find $\lim\limits_{n \to \infty} \frac{1}{n}\sum\limits^{2n}_{r =1} \frac{r}{\sqrt{n^2+r^2}}$

Find $$\lim_{n \to \infty} \frac{1}{n}\sum^{2n}_{r =1} \frac{r}{\sqrt{n^2+r^2}}$$ My approach : $$\lim_{n \to \infty} \frac{1}{n}\sum^{2n}_{r =1} \frac{r}{\sqrt{n^2+r^2}} =\lim_{n \to \infty} ...
0
votes
1answer
106 views

Show:$ \frac 12 n (n+1)=(n+1)(\frac 12 n+1)=\frac 12 (n+1)(n+2)$

Given: $$ \sum_{k=1}^n k=1+2+3+\dots +n=\frac 12 n (n+1)$$ Show: $$ 1+2+3+\dots +n+(n+1)=(n+1)(\frac 12 n+1)=\frac 12 (n+1)(n+2)$$ The following is me wallowing in ignorance and failing to derive ...
1
vote
1answer
60 views

Limit of the set $(f(ax),f(x)]$ as $x\to\infty$ and $f$ non-increasing

Assume $f>0$ is non-increasing with $\lim_{x\to\infty}f(x)=0$ and $a\in(1,\infty)$. I assume that $$\limsup_{x\to\infty} (f(ax),f(x)] =\emptyset.$$ How can I prove it? I know that I have to show ...
1
vote
2answers
94 views

What is the sum of this by telescopic method?

$$\sum_{r=1}^\infty \frac{r^3+(r^2+1)^2}{(r^4+r^2+1)(r^2+r)}$$ This is to be done by telescopic method. I've used many things like writing $r^4+r^2+1=(r^2+r+1)(r^2-r+1)$ but have failed. I think all ...
37
votes
1answer
798 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
0
votes
1answer
163 views

Matrix Exponential equality

I was reading about the matrix exponential function and I came across this: If $xy = yx$ then $$ \exp(x+y) = \exp(x)\cdot\exp(y) $$ My textbook gives a proof as follows: $$ \exp(x+y) = ...
0
votes
1answer
209 views

Finding the $n^{\text{th}}$ term of $\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\ldots$

I need help on finding the $n^{\text{th}}$ term of this infinite series? $$ s=\frac{1}{4}+\frac{1\cdot 3}{4\cdot 6}+\frac{1\cdot 3\cdot 5}{4\cdot 6\cdot 8}+\ldots $$ Could you help me in writing the ...
0
votes
2answers
96 views

A result about infinite series: How to prove this?

Let $\{a_n\}$ be a sequence of positive real numbers such that, for some $N \geq 1$, some $s>1$, and some $M>0$, we have $$ \frac{a_{n+1}}{a_n} = 1 - \frac{A}{n} + \frac{f(n)}{n^s} $$ for all ...
7
votes
1answer
227 views

Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$

Find the sum of the following : (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$ I tried : ...
6
votes
3answers
258 views

Does the series $1 + \frac{1}{2} - \frac{1}{3} + \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \dots$ converge?

Does the following variant of the harmonic series converge? If it diverges (which I think it does), can I know if it diverges to $\infty$ or has no limit? Note that the series is not alternating in ...
1
vote
0answers
66 views

looking for explanation behind solution for a 1st order recurrence relation.

In lecture, we covered 1st order recurrence relations and came up with a solution by inspection. I sort of see that we're finding the next term in the sequence by multiplying the initial condition by ...
6
votes
1answer
158 views

Intruiging Symmetric harmonic sum $\sum_{n\geq 1} \frac{H^{(k)}_n}{n^k}\, = \frac{\zeta{(2k)}+\zeta^{2}(k)}{2}$

I proved the following equation $$\sum_{n\geq 1} \frac{H^{(k)}_n}{n^k}\, = \frac{\zeta{(2k)}+\zeta^{2}(k)}{2}$$ We define $$H^{(k)}_n=\sum_{m= 1}^n \frac{1}{m^k}$$ I am looking forward to ...
4
votes
2answers
280 views

Solve : $\frac{2012!}{2^{2010}}-\sum^{2010}_{k=1} \frac{k^2k!}{2^k}-\sum^{2010}_{k=1} \frac{k\cdot k!}{2^k}$

Solve : $$\frac{2012!}{2^{2010}}-\sum^{2010}_{k=1} \frac{k^2k!}{2^k}-\sum^{2010}_{k=1} \frac{k\cdot k!}{2^k}$$ Can we take like this : Let us take (k+1)th term = $$\frac{(k+1)^2(k+1)!}{2^{k+1}} ; ...
5
votes
3answers
256 views

Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
9
votes
4answers
311 views

What is the product of this by telescopic method?

$$\prod_{k=0}^{\infty} \biggl(1+ {\frac{1}{2^{2^k}}}\biggr)$$ My teacher gave me this question and said that this is easy only if it strikes the minute you read it. But I'm still thinking. Help! ...
3
votes
2answers
355 views

Convergent Sequence and Cauchy Criterion- Counter Example

Consider the sequence $\left \{ x_{n} \right \}$ that satisfies the condition: $$\left | x_{n+1}-x_{n} \right |< \frac{1}{2^{n}} \ \ \ for\ all\ n=1,2,3,...$$ Part (1): Prove that the sequence ...
0
votes
1answer
63 views

On a infinite series problem of IMC

In the solution 2 of problem of 2 of IMC 1999 I want to ask why $$\sum_{n=1}^{\infty}\frac{\pi (n)}{n^2}= \sum_{n=1}^{\infty}(\pi (1)+ \pi(2)+\cdots + \pi(n))\left( ...
0
votes
2answers
104 views

Find n term of sequence

A sequence is given: $$1,10,11,100,101,110,111,1000,\dots,a_n,\dots$$ The question is: what is the value of $a_n$ for a given $n$? I have tried a lot of patterns but was not able to meet the ...
5
votes
2answers
124 views

How to find the sum $\displaystyle\sum^n_{k=1} (k^2+k+1)k!$

How to find the sum $$\sum^n_{k=1} (k^2+k+1)k!$$ What I tried as follows : $$\sum^n_{k=1} (k^2+k+1)k!$$ =$$\sum^n_{k=1} (k^2)k!+ \sum^n_{k=1} (k)k! + \sum^n_{k=1} (1)k!$$ Now we can write ...