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

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1
vote
0answers
64 views
+50

Inequality involving a strictly positive sequence

I was asked to prove the following : \begin{array}{l} x_n > 0,\,\forall n \in Z^ + \\ \lim \sup (\frac{{x_{n + 1} + x_1 }}{{x_n }})^n \ge e \\ \end{array} Is my approach correct ? Using ...
15
votes
4answers
445 views
+50

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for ...
5
votes
1answer
199 views
+100

Compute limit of the sequence $x_n$

Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
6
votes
0answers
231 views
+100

It is easy to show that $\displaystyle S_m=\sum_{n=1}^\infty \frac{n}{2^n + m}$ converges for any natural$\ m$, but what is its value?

In fact the series would converge even if$\ m$ were not natural, I just wanted to state that it is natural in my case. I have found the partial sum formula of$\ S_0$,$\displaystyle \sum_{n=1}^k ...
2
votes
0answers
156 views
+200

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
4
votes
2answers
98 views
+50

Closed form of $\sum_{k=0}^{\infty} \frac{k^a\,b^k}{k!}$

While working on this question I think I've found a closed-form expression for the following series, but I don't know how to prove it. Let $a \in \mathbb{N}$ and $b \in \mathbb{R}$. Then ...
1
vote
0answers
39 views
+50

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=(((√3+2)^{2^{p-1}}+1)/((2^{p}-1)(√3+2)^{2^{p-2}}))$$ is a natural number then $2^{p}-1$ is a prime number. ...
0
votes
0answers
16 views
+50

Concerning the classical normalized Eisenstein series

Earlier I asked this question. As of today, it has not been answered. Yet still, I have a follow-up question: In general, how does one express $E_4(\tau)$ and $E_6(\tau)$ in closed form for special ...