# Questions tagged [sequences-and-series]

For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.

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### Are there bounded monotone sequences whose product is not monotone?

I was wondering whether or not the product of two monotone and bounded sequences must be monotone. I can think of e.g. $(x_n)=(-1,-1,1,1,1,…)$ and $(y_n)=(1,0,-1,-1,-1,…)$ that are clearly both ...
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I'm trying to determine whether this series is convergent or divergent: $$\sum_1^\infty \frac{e^{1/n}-1}{n}$$ I thought directly that $e^{1/n} \geq 1$ and therefore $\frac{e^{1/n}-1}{n} \geq \frac{... 6 votes 2 answers 121 views ### Check if this series is convergent or not I've been going crazy for a long time in determining if this series converges or diverges. Most likely it converges.$\sum_{n=1}^\infty (\frac{n}{2} sin\frac{1}{n})^\frac{n^2+1}{n+2}$I am stuck in ... 7 votes 1 answer 306 views ### Is there a connection between the sum of$\sum_{n=1}^\infty \frac{\ln(n+1)-\ln(n)}{n}$and the Riemann zeta function? It can be shown that for every integer$p\geq 0, this integral identity holds: \begin{align} \int_1^\infty\frac{1}{x^{p+2}\lfloor x\rfloor}\text{ }dx &= -1+\frac{1}{p+1}\sum_{m=2}^{p+2}\zeta(m)\\ ... 3 votes 2 answers 189 views ### How to tackle the integral\int_{0}^{\infty} \frac{\ln x}{x^{n}-1} d x$? In my post, I started to investigate the integral$\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x$. Fortunately, $$\displaystyle \int_{0}^{\infty} \frac{\ln x}{x^{2}-1} d x =2 \int_{0}^{1} ... 2 votes 1 answer 32 views ### If the quotient of limits converge to 0 then there exists M \in \mathbb{N} such that for all n \ge M, y_n > |x_n|. Suppose \{x_n\} and \{y_n\} are sequences (not necessarily convergent) such that y_n > 0 for all n \in \mathbb{N} and$$\lim_{n \rightarrow \infty} \frac{x_n}{y_n} = 0$$Prove that there ... 1 vote 2 answers 36 views ### Prove \sum_{x=1}^\infty xr^x = {r\over r-1} by taking the limit of the partial sum formula There are multiple ways to derive \sum_{x=1}^\infty xr^x = {r\over (r-1)^2} mentioned here How can I evaluate \sum_{n=0}^\infty(n+1)x^n? but none of them show the derivation by taking the limit of ... -1 votes 1 answer 55 views ### How can I Taylor expand an expression that has a square root under exponent of -1/2? So, in a quantum mechanics homework I have, I have an eigenstate |\psi\rangle given by$$|\psi\rangle=\left(1+\frac{\Omega^2}{\left(\delta+\sqrt{1+\left(\frac{\Omega}{\delta}\right)^2}\right)^2} \... 2 votes 1 answer 81 views ### Closed-form or integral representation for$\sum_{n=0}^\infty\frac{e^{-\lambda}\lambda^n}{n!}\phi(z-n;\mu;\sigma)$. Let$Z=X+Y$, where$X\sim\operatorname{Poisson}(\lambda)$and$Y\sim\mathcal N(\mu,\sigma^2)$are independent. The density of$Z$can thus be expressed as an infinite component Gaussian mixture of ... 0 votes 1 answer 30 views ### Finding the cluster points of a sequence in$\mathbb{R}^3\$

I haven't found many examples of how to find cluster points for sequences of real numbers instead of sets and for some reason I've found more confusing to do this with sequences than with sets. ...
It is given that $$L=\lim _{k \rightarrow \infty}\left\{\frac{e^{\frac{1}{k}}+2 e^{\frac{2}{k}}+3 e^{\frac{3}{k}}+\cdots+k e^{\frac{k}{k}}}{k^2}\right\}$$ I tried solving it but I am stuck on this, ...