Articles of sequences and series

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent?

Is $\sum_{n=1}^\infty {1\over 3^{\sqrt{n}}}$ convergent ? I use it to compare with $1/n^2$, and then I used LHôpitals rule multiple times. Finally , I can solve it. However,I think we have other methods! somebody can help me?

Why does $\sum_{n=1}^{\infty}\frac{\cos\frac{1}{n}}{n}$ diverge but $\sum_{n=1}^{\infty}\frac{\sin\frac{1}{n}}{n}$ converges?

Why does $\sum_{n=1}^{\infty}\frac{\cos\frac{1}{n}}{n}$(I’m told it can be compared to the harmonic sequence, but I don’t see the tangible comparison) diverge but $\sum_{n=1}^{\infty}\frac{\sin\frac{1}{n}}{n}$ converges?

What is the most complete book of integrals and series?

I’m looking for something like “If it’s not in this book, it’s not known”. I’ve got a copy of Gradshteyn and Ryzhik, which seems pretty good. But I’m hoping there are some better ones out there.

Inequality of Finite Harmonic Series

I’m asked to prove that for $n\in \mathbb{N}$ $$\frac{1}{1} + \frac{1}{2} +\cdots+\frac{1}{n} \geq 1 + \frac{n}{2}$$ by induction. I’ve got a feeling that the problem isn’t right (since it isn’t true for any $n\in \mathbb{N}$), does anyone know the result and what it should be? Edit: The author says that the result can be used […]

The value of $\sum_{k=0}^\infty k^n x^k$

This question already has an answer here: Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$? 4 answers

Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} – 1\right)\,$ converge?

I’m trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart’s Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} – 1)$$ I’ve considered all of the techniques I know (integration test, ratio test, etc.), but I upon inspection, none of them will solve this problem. Any hints or help […]

Given $a_1,a_{100}, a_i=a_{i-1}a_{i+1}$, what's $a_1+a_2$?

I’ve been given the following puzzle Let $a_1, a_{100}$ be given real numbers. Let $a_i=a_{i-1}a_{i+1}$ for $2\leq i \leq 99$. Further suppose that the product of the first $50$ is $27$, and the product of all the $100$ numbers is also $27$. Find $a_1+a_2$. I tried the following, looking at the sequence for a moment […]

Taylor expansion of $\frac{1}{1+x^{2}}$ at $0.$

I am trying to find the Taylor expansion for the function $$f(x) = \frac{1}{1+x^{2}}$$ at $a=0.$ I have looked up the Taylor expansion and concluded that it would be sufficient to show that \begin{eqnarray} f^{(2k)}(0) & = & (-1)^{k}(2k)! \\ f^{(2k+1)}(0) & = & 0 \end{eqnarray} The problem I am having is that the derivatives […]

Show that $f(x)=\begin{cases}1&\text{if }x\in\mathbb{Q}\\0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q}\end{cases}$ is discontinuous everywhere

Can any one help me to answer this question: Show that $$f(x)=\begin{cases} 1 &\text{if }x\in \mathbb{Q} \\ 0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q} \end{cases}$$ is discontinuous everywhere. Notice: use this theorem Let $f:D\rightarrow \mathbb{R}$ and let $c\in D$. Then $f$ is continous at $c$ if and only if, whenever $X_n$ is a sequence in $D$ that converges […]

Estimate a sum of products

Let $0 < \alpha < 1$. Show that for $\lambda > 0$ big enough $$\sum_{n=1}^\infty \prod_{k=1}^n \frac{1}{\lambda k^{-\alpha} + 1} < \infty$$ I think $\lambda = 1$ is enough. You could then use the estimation \begin{align*} \prod_{k=1}^n \frac{1}{ k^{-\alpha} + 1} &\le \prod_{k=1}^n \frac{1}{ n^{-\alpha} + 1} \\ &= \left(\frac{1}{ n^{-\alpha} + 1}\right)^n \\ &= […]