New posts in sequences-and-series

Can one show that $\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$ without using the Euler-Maclaurin formula?

sequences-and-series inequality harmonic-numbers

Complex Analysis Solution to the Basel Problem ($\sum_{k=1}^\infty \frac{1}{k^2}$) [duplicate]

sequences-and-series complex-analysis complex-numbers pi

Proof that if $a_1=1$ and $a_{n+1}=1+\frac{1}{1+a_n}$

real-analysis sequences-and-series recurrence-relations

How can I prove $\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$

sequences-and-series complex-analysis

Finding the value of the infinite sum $1 -\frac{1}{4} + \frac{1}{7} - \frac{1}{10} + \frac{1}{13} - \frac{1}{16} + \frac{1}{19} + ... $ [duplicate]

calculus integration sequences-and-series limits closed-form

Stuck at proving whether the sequence is convergent or not

real-analysis sequences-and-series trigonometry convergence-divergence limsup-and-liminf

Evaluating $\sum_{n=0}^{\infty}{\sin^{3}\left(3^{n}\right) \over 3^{n}}$

sequences-and-series trigonometry

Evaluate of $\lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n}$

calculus real-analysis sequences-and-series limits convergence-divergence

Find $\sum_{n=0}^\infty\frac{2^n}{3^{2^{n-1}}+1}$

sequences-and-series

Is collapsing considered a legitimate proof?

sequences-and-series proof-writing

$\sum\limits_{k=1}^{\infty} {1 \over k}{1 \over 2k-1}$ how to show that this is $ 2 \ln 2 $?

number-theory sequences-and-series

Explain why $\int_0^\infty\frac{\sin{4x}}{x}\prod\limits_{k=1}^n \cos\left(\frac{x}{k}\right) dx\approx\frac{\pi}{2}$

calculus integration sequences-and-series pi trigonometric-integrals

Find $\sum_{n=1}^\infty\frac{2^{f(n)}+2^{-f(n)}}{2^n}$, where $f(n)=\left[\sqrt n +\frac 12\right]$ denotes greatest integer function

sequences-and-series summation ceiling-and-floor-functions

Find all limit points of $M=\left \{ \frac{1}{n}+\frac{1}{m}+\frac{1}{k} : m,n,k \in \mathbb{N} \right \}$ in space $(M,\rho_{e})$

general-topology sequences-and-series limits

Infinite sum of Bessel Functions

sequences-and-series special-functions

Is there a sequence with an uncountable number of accumulation points?

real-analysis sequences-and-series general-topology

Limit $\lim_{x\to\infty}\left(\sum_{n=1}^{\infty}\left(\frac{x}{n}\right)^n \right)^{1/x}$

sequences-and-series limits

Find the limit $\lim_{n\to\infty}\left(\sqrt{n^2+n+1}-\left\lfloor\sqrt{n^2+n+1}\right\rfloor\right)$ [duplicate]

sequences-and-series limits radicals integers ceiling-and-floor-functions

Calculate $\frac13+\frac29+\frac{1}{27}+\frac{2}{81}+\frac{1}{243}+\frac{2}{729}+\ldots$

sequences-and-series

For $a_n>0$ such that $\sum a_n $ converges, show that there exist $c_n>0$ such that $c_n\to \infty$ and $\sum a_n c_n$ is finite.

real-analysis sequences-and-series