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

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0
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0answers
21 views

What is the infinite sum of $a^{b^x}$?

What would $$\sum^{\infty}_{i=0}(1/2)^{4^n}$$ be and how to determine it?
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1answer
7 views

In an irreducible, aperiodic, null-recurrent Markov chain holds $\sum_n p_{ij}^{(n)} = \infty$

My lecture notes state the following theorem: Theorem 2. Let $(X_n)$ be an irreducible, aperiodic, null-recurrent Markov chain. Then $$\forall i,j \in S : p_{ij}^{(n)}\to 0 \text{ and } \sum_n ...
3
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2answers
28 views

Sequence Convergence using bounding sequences

Can someone help me? The hint they give me is to find two bounding sequence, but I don't understand how this could help me Thanks
4
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2answers
79 views

Challenging Infinite summation involving the zeta function [duplicate]

Evaluate: $$\large\sum_{k=1}^{\infty}\left(\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}\right)^2$$ MY ATTEMPT: Recognizing that $\zeta{(2)}-\sum_{n=1}^{k}\frac{1}{n^2}$ can be written as $ ...
3
votes
5answers
75 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
1
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1answer
37 views

Viewing a sequence as a function on the space of positive integers

I see the following lines in a book : " Consider a bounded sequence of real or complex numbers $\{\eta_n\}$. Such a sequence $\{\eta_n\}$ defines a function $x(n) = \{\eta_n\}$ defined on the ...
1
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2answers
29 views

Question about point-wise convergent sequence of functions.

Let $$f_n(x)=n^2x(1-x^2)^n$$ be a sequence of functions on $[0,1]$. For $x=0$ and $x=1,$ clearly $f_n(x)=0$. Also for any $x_0$ in the open interval $(0,1)$, we have $0<1-x_0^2<1$. Therefore $$ ...
5
votes
2answers
79 views

Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$

I am trying to evaluate the finite sum \begin{equation} f(u)=\sum_{k=0}^n{n\choose k}^2u^k,\quad 0<u\le1 \end{equation} In an first attempt, I think of the generating function \begin{equation} ...
1
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2answers
38 views

Proper way to express 0 in this case?

If 0=(x-a)(x-b)(x-c)...(x-x)..=0. So it's a product sum that we write with pi instead of sigma but how? There should be indexes but I'm not convinced that I understand what notation to use. $$\prod_{ ...
2
votes
1answer
36 views

For any series that diverges, does there exist a sequence that converges to 0 yet the product diverges

Suppose $\sum_{n=1}^{\infty} a_n = \infty$. Does there exists a sequence $\{b_n\}$ such that $ \lim_{n \rightarrow \infty} b_n = 0$ where $\sum_{n=1}^{\infty} a_n b_n = \infty$? I am able to prove ...
3
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0answers
47 views

Finding a formula for a given series. [duplicate]

I'm having trouble figuring out how to evaluate: $$\sum _{j = 1}^{n} j!j$$ I've tried plugging in numbers and looking for a pattern and I've also tried to find a general form like: $$n=1:$$ $$ 1*1 ...
5
votes
4answers
106 views

Limit of a sequence of products

How do you prove the following? $$\lim_{n\,\to\,\infty}\,\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\ =\ 0$$
2
votes
1answer
24 views

Proof of absolute convergence [duplicate]

I am independently studying calculus using MIT's publicly available materials on OCW. One of the Final Exam practice question is the following: Suppose the series $\sum_{n=1}^{\infty}a_n$ converges ...
4
votes
3answers
68 views

How to derive the closed form of the sum of $kr^k$

$$ \sum_{k=0}^{n}kr^k = r\frac{1-(n+1)r^n + nr^{n+1}}{ (1 - r)^2 } $$ How to derive it? I read about some finite calculus, and i understand how to tackle sums of $x^2$, $x^3$, etc.. But I don't know ...
3
votes
4answers
236 views

Sequence proof (by induction, presumably) giving me trouble.

Let $a_1,...,a_n$ be a sequence of positive numbers. Show that $$(a_1+a_2+\cdots+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}\right)\geq n^2$$ Hint: Use the fact that for $x>0$ we ...
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2answers
32 views

How to take the limit of the improper integral of a sequence of functions

Suppose $f_1, f_2, . . .$ are (Riemann) integrable functions. Then what is the $\epsilon$ definition of $$\lim_{n \rightarrow \infty} \lim_{M \rightarrow \infty} \int_{0}^{M} f_n(x) dx = L $$ for $L ...
4
votes
2answers
115 views

Is there any closed form for the finite sum $1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+…+\dfrac{1}{n}?$ [duplicate]

I know that the infinite summation $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}+...$$ is divergent and also the sequence $$1+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{n}-\ln ...
0
votes
0answers
19 views

Estimate on the difference of quotients

The following is supposedly true (found it in a paper), however I fail to see why. Let $L(x)$ be a function that goes to $0$ as $x\rightarrow\infty$, $g(n)$ a sequence which goes to $\infty$ as ...
0
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1answer
13 views

Showing that two infinite series converge to the same value

I was preparing for my exam and came across this problem. Show that The series on the left hand side is the power series of $\ln(1+x)$ evaluated at $x=1$. This is what i've done so far. From ...
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1answer
28 views

Exercise 4.4, Mathematical Analysis 2nd ed. - Apostol

Two sequences of positive integers $\{a_n\}$ and $\{b_n\}$ are defined recursively by taking $a_1=b_1=1$ and equating rational and irrational parts in the following equation ...
0
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1answer
20 views

A question about the relation between divergence and absolute divergence.

Princeton Lectures in Complex Analysis by Stein and Shakarchi says the following: If $|z| > R$, then a similar argument proves that there exists a sequence of terms in the series whose ...
0
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2answers
57 views

Direct evaluation of a series from Euler's identity.

Is there a direct way to evaluate: $$ \sum_{k=0}^{\infty} (-1)^k \dfrac{\pi^{2k}}{(2k)!}=-1 $$ Note that this follows from Euler's identity.
1
vote
1answer
39 views

If $(c_n)_n$ is the sum of geometric and arithmetic sequences. How to get the original sequences back?

If we have a geometric sequence $a= (a_n)_n$ and an arithmetic sequence $b=(b_m)_m$. We can find the $n$th term of $a+b$ easily. Now, suppose we have a sum of geometric sequnce and arithmetic ...
3
votes
1answer
67 views

Consecutive Prime Problem

Consecutive primes whose quotient of their product and sum is itself a prime number. $$ 2 \times 3 \times 5 = 30 $$ $$ 30/10 = 3 $$ $$ 3 \times 5 \times 7 = 105 $$ $$ 105/15 = 7 $$ Question: ...
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0answers
23 views

Series of multi sets

Given that a set of numbers $K = \{n_1, n_2, n_3, ... \}$. Multiple subsets are formed by randomly extracted numbers from $K$. Then series are formed by extracting numbers from the subset orderly. ...
1
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2answers
100 views

Proof that $\dfrac{1}{e^x}=e^{-x}$ without converting it to $e^{x}e^{-x}=1$.

I want to show that $\dfrac{1}{e^x} = e^{-x}$ from the Taylor expansion of $e^x$. To express $\dfrac{1}{e^x}$ as a power series, I let: $$ \left(\dfrac{1}{0!}x^0 + \dfrac{1}{1!}x^1 + ...
0
votes
1answer
44 views

How to evaluate $\sum_{k=0}^{n} \alpha^k \binom{n}{k}$?

I am trying to show that the function that satisfies $f^\prime(x)=f(x)$ with $f(0)=1$ behaves in an exponential way (in other words, I want to justify writing it as $e^x$). I need to show that: $$ ...
9
votes
4answers
247 views

What's the sum of this series? [duplicate]

I would like to know how to find out the sum of this series: $$1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \frac{1}{5^2} - \frac{1}{6^2} + \cdots$$ The answer is that it converges to a sum ...
3
votes
2answers
81 views

$\sum \tan ( 1/n)$ diverges

Show that the series $$\sum_n \tan\left(\frac{1}{n}\right)$$ diverges. I dont have any attempt to do, since I am having some troubles with series including geometric functions. I would be glad if I ...
7
votes
2answers
133 views

Summing various rearrangements of $1-\frac12+\frac13-\frac14+\cdots$

Show that the series $1-\frac12+\frac13-\frac14+\ldots$ converges to $\log2$ but the rearranged series: ...
2
votes
3answers
64 views

Sequence $x_n= \frac{a^n + b^n}{(ab)^n}$

Verify if the sequence $$x_n=\frac{a^n + b^n}{(ab)^n}$$ converges, assuming $ab>1$ and $a,b>0$. My attempt: If $ a+b<ab$, then the sequence is convergent, since, if we consider the series ...
1
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0answers
45 views

Is there a closed-form expression for this matrix power series?

I encountered a matrix power series: $$ X = M + PMP^{t} + P^{2}M(P^{t})^{2} + \cdots, $$ where $M$ is a real symmetric matrix, and $P$ is a real square matrix. Assuming that this series converges, ...
4
votes
1answer
64 views

Are there other methods to evaluate $\frac{1^{-4}+2^{-4}+3^{-4}+4^{-4}+\cdots}{1^{-4}+3^{-4}+5^{-4}+7^{-4}+\cdots}$?

Are there other methods to evaluate the following series? $$\frac{1^{-4}+2^{-4}+3^{-4}+4^{-4}+\cdots}{1^{-4}+3^{-4}+5^{-4}+7^{-4}+\cdots}$$ My attempt is as follows, \begin{align} ...
2
votes
1answer
30 views

Convergence study of a series of functions

I am studying the convergence of the series $$ \sum_{n=0}^{\infty}\frac{\sin (x^n)}{(1+x)^n} $$ where $x \in \mathbb R$. My initial approach was to use the ratio test, but I am not getting to ...
0
votes
1answer
52 views

Summation of series of $2/(r-1)(r+1)$ using the method of differences

Verify the identity $$\frac{2r-1}{r(r-1)}-\frac{2r+1}{r(r+1)}=\frac{2}{(r-1)(r+1)}$$ Hence, using the method of differences, prove that ...
4
votes
0answers
56 views

Limit of infinite product

Is it possible to find an analytic form for the limit of the infinite product: $$ \prod_{n=1}^\infty\frac{1+x^{\delta^n}}{2} $$ where $ x>0 $ and $0<\delta<1$?
0
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3answers
44 views

Convergence of series $\sum$$u_n$= $\sum$$\frac{n! x^n}{(n+1)^n}$

My series is $$1+\frac{x}{2}+\frac{2! x^2}{3^2}+\frac{3!x^3}{4^3}+\ldots$$ My approach: $$u_n= \frac{n! x^n}{(n+1)^n}$$ So, $$u_{n+1}= \frac{(n+1)! x^{n+1}}{(n+2)^{n+1}}$$ So, ...
2
votes
1answer
15 views

Limit of sum of series

Let $$\sum_{i=1}^\infty c_i^{(0)}$$ be a positive convergent series. Then the series: $$\sum_{i=1}^\infty c_i^{(1)}=\sum_{i=1}^\infty \ln(1+c_i^{(0)})$$ is also a positive convergent series. ...
1
vote
1answer
50 views

Test for convergence of the series.

$$\frac{1}{1 \cdot 3}\ + \frac{2}{3 \cdot 5}\ + \frac{3}{5 \cdot 7}\ + \cdots$$ What is the $nth$ term here and what test should I use?
0
votes
1answer
25 views

How to get this inequality

Let $c>0$, $n \in \mathbb N$ and $q>1$. How to get the following approximating inequality when $n$ is large, please? To be more specific, I cannot see how to get rid of the square root. $$ ...
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votes
1answer
34 views

Sequence converging to the supremum. [on hold]

let $T\subset \mathbb R$ let $a=\sup(T)$, and suppose that and $a<\infty$ and that $a \notin T$. Show that there exist a sequence $(a_n)\subset T$ for which $\lim_{n\to\infty}a_n=a$. If you are ...
0
votes
2answers
34 views

Determining if series converges or diverges

The Series is For this series the ratio test is inconclusive. I have rewritten the series as Currently i am approaching the problem using limit test. I couldn't progress from this point. Any ...
1
vote
1answer
26 views

What kind of convergence is $\sum |f_n|$?

We know that $\sum |f_n|$ converges on $E\subseteq \mathbb R$ then $\sum f_n$ is said to converge absolutely on $E$. But in terms of pointwise or uniform convergence, I am willing to know what kind ...
0
votes
2answers
22 views

Serie $\sum \frac{\cos n -\sin n\pi}{n}$

How to show that the serie $$\sum \frac{\cos n -\sin n\pi}{n}$$ Diverges? In general, I am having troubles with the series that include trigonometric functions, but I think there is no general ...
0
votes
0answers
9 views

Calculate and Forecast Churn Rate of Members

I need to estimate the total number of times users logged in to an app, cumulatively at the end of a year. If I have a mobile app, and 100,000 people download it. And they only use it once, then the ...
1
vote
2answers
38 views

Naive proof that $\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$ [duplicate]

As part of a larger proof, I must show that: $$\sum_{n=1}^{N-1}\cos(2\pi\frac{n}{N})=-1$$ I have thought about this but can't figure out how to get my hands on the value since I don't know any ...
0
votes
2answers
57 views

A identity relating a infinite series and a definite integral [duplicate]

Prove that, $$ \sum_{n=1}^{\infty} \frac{1}{n^n} = \int_{0}^{1} x^{-x}dx$$ I made no significant progress, I'm looking for hint/ideas to approach this problem. Thanks!
0
votes
3answers
43 views

Is this positive term series convergent?

My series is: $\frac{1}{1+2^{-1}}\ +\frac{1}{1+2^{-2}}\ +\ldots$ I see my $nth$ term is $\frac{1}{1+2^{-n}}$ How do I test for its convergence?
1
vote
2answers
30 views

sequence: use of stirling formula

I want to use the Sterling formula which says that: $lim_{n \rightarrow \infty}\dfrac{n!}{\sqrt{2*\pi}n^{n+1/2}*e^{-n}}=1$ I want to use it to show that $\lim_{n \rightarrow ...
1
vote
1answer
23 views

Test for convergence of positive term series

$$\frac{2}{1^p}\ + \frac{3}{2^p}\ + \frac{4}{3^p}\ +\ldots\,.$$ I can see that the $nth$ term is $\frac{n+1}{n^p}$ How do I test for its convergence?