Articles of sequences and series

How to evaluate $\int_0^x\vartheta_3(0,t)\ dt$?

Due to this question, I found myself trying to take the following integral: $$\int_0^x\vartheta_3(0,t)\ dt=\ ?$$ However, I know not of how to do this. As per this post, I find the evaluation at $x=1$ to be $\frac\pi{\tanh\pi}$. It is equivalent to trying to evaluate the following series: $$\sum_{n=0}^\infty\frac{x^{n^2}}{n^2+1}$$ My end problem is that I […]

Proving $\sum_{n \geq 1} \frac{a_n}{1+ a_n}$ diverges provided $\sum a_n$ diverges and $(a_n)$ is decreasing and nonegative

Could someone tell me why this might a wrong solution? The solution looked nothing like mine Let $t_n$ be the partial sums of $\sum_{n \geq 1} \frac{a_n}{1+ a_n}$ $$t_n = (1-\frac{1}{1+a_1}) + (1-\frac{1}{1+a_2}) + \dots + (1-\frac{1}{1+a_n}) \\ \geq (1-\frac{1}{1+a_n}) + (1-\frac{1}{1+a_n}) + \dots + (1-\frac{1}{1+a_n}) \\ \geq n – \frac{n}{1+a_n} \geq n$$ So $n […]

Show that this limit is positive,

I want to show that, for $\alpha = \frac{1}{2}$, $$\lim_{x \to 0} \sum_{n=1}^{\infty}\frac{x}{(1+nx^2)n^{\alpha}}>0$$ Any ideas are welcome. (In a previous question, I considered the case for $\alpha > \frac{1}{2}$, from which we were able to derive a uniform upper bound and use Weierstrass M-test and Dominated Convergence Theorem to establish the continuity of the series […]

Smallest function whose inverse converges

Is there some increasing function $f(n)$ that grows slower than $n^{c}$ for some $c > 1$ such that $\sum_{n=1}^{\infty} \frac{1}{f(n)}$ converges?

Prove if $a>1$ then $\lim_{n\rightarrow\infty}a^{n}=\infty $

Good morning i was thinking about this problem and I make this. I need someone review my exercise and say me if that good or bad. Thank! Problem: Prove if $a>1$ then $\lim_{n\rightarrow\infty}a^{n}=\infty $ Proof: Suppose $\left\{ a^{n}\right\} $ is monotonically increasing. In other words $a^{n}<a^{n+1}< a^{n+2}…$ and Suppose $\left\{ a^{n}\right\} $ is Bounded set […]

Sum of real powers: $\sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$

Let $\{x_i\}_{i=1}^{N}$ be positive real numbers and $\beta \in \mathbb{R}$. Can we say that: $$ \sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$$ I know that this holds if $\beta \in \mathbb{N}$. Does the above inequality have a name in case it’s true?

How to bound the truncation error for a Taylor polynomial approximation of $\tan(x)$

I am playing with Taylor series! I want to go beyond the basic text book examples ($\sin(x)$, $\cos(x)$, $\exp(x)$, $\ln(x)$, etc.) and try something different to improve my understanding. So I decided to write a program for approximating $\tan(x)$. But I am having difficulty. I want to use the Taylor series of $\tan(x)$ to approximate […]

Limit of $a_n = \sum\limits_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$

Given $a_n = \sum\limits_{k=1}^{n} \left(\sqrt{1+\frac{k}{n^2}}-1\right)$, find $\lim\limits_{n \to \infty} a_n$. My try: To simplify, $$a_n = \frac{\displaystyle\sum_{k=1}^{n}\sqrt{n^2+k}-n}{n}$$ and I’m stuck from there. In addition, I have made a program to find the limit, which says it’s 1/4. Can anybody give me a hint to start? Thanks for your time!

Closed form of $ x^3\sum_{k=0}^{\infty} \frac{(-x^4)^k}{(4k+3)(1+2k)!} $

I was trying to find a closed form of $$ f(x) = x^3\sum_{k=0}^{\infty} \frac{(-x^4)^k}{(4k+3)(2k+1)!} $$ $f(x)$ converges for all $x$ by the ratio test. I began by making the it look like a power series and then differentiating $$ f(x) = \sum_{k=0}^{\infty} \frac{(-1)^kx^{4k+3}}{(4k+3)(2k+1)!} $$ $$\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} \sum_{k=0}^{\infty} \frac{(-1)^kx^{4k+3}}{(4k+3)(2k+1)!} $$ $$ \frac{\partial […]

Easy way to find out limit of $a_n = \left (1+\frac{1}{n^2} \right )^n$ for $n \rightarrow \infty$?

What’s an easy way to find out the limit of $a_n = \left (1+\frac{1}{n^2} \right )^n$ for $n \rightarrow \infty$? I don’t think binomial expansion like with $\left (1-\frac{1}{n^2} \right )^n = \left (1+\frac{1}{n} \right )^n \cdot \left (1-\frac{1}{n} \right )^n$ is possible. And Bernoulli’s inequality only shows $\left (1+\frac{1}{n^2} \right )^n \geq 1 + […]