Articles of sequences and series

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}… =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: $$\displaystyle T = \left( \begin{array}{ccccccc} +0&+0&+0&+0&+0&+0&+0&\cdots \\ +1&-1&+1&-1&+1&-1&+1 \\ +1&+1&-2&+1&+1&-2&+1 \\ +1&+1&+1&-3&+1&+1&+1 \\ +1&+1&+1&+1&-4&+1&+1 \\ +1&+1&+1&+1&+1&-5&+1 \\ +1&+1&+1&+1&+1&+1&-6 \\ \vdots&&&&&&&\ddots \end{array} \right)$$ Is it true that $\displaystyle \log(n)=\sum\limits_{k=1}^{\infty}\frac{T(n,k)}{k}$$\;$?

Uniform limit of holomorphic functions

Let $\{f_n\}$ be a sequence of holomorphic functions defined in a generic domain $D \subset \mathbb{C}$. Suppose that there exist $f$ such that $f_n \to f$ uniformly. My question is: is it true that $f$ is holomorphic too?

How to prove $\sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}} = \frac{2+\sqrt 5 +\sqrt{15-6\sqrt 5}}{2}$

Ramanujan stated this radical in his lost notebook: $$\sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}} = \frac{2+\sqrt 5 +\sqrt{15-6\sqrt 5}}{2}$$ I don’t have any idea on how to prove this. Any help appreciated. Thanks.

If $\{x_n\}$ satisfies that $x_{n+1} – x_n$ goes to $0$, is $\{x_n\}$ a Cauchy sequence?

Since the definition of Cauchy sequence is: Understanding the definition of Cauchy sequence, I noticed we need an absolute value for $a_m-a_n$ in the definition so the statement would be false. But I can’t find such a couterexample. Maybe it is true?

Can we add an uncountable number of positive elements, and can this sum be finite?

Can we add an uncountable number of positive elements, and can this sum be finite? I always have trouble understanding mathematical operations when dealing with an uncountable number of elements. Any help would be great.

How to explain the formula for the sum of a geometric series without calculus?

How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that $$1 + \frac{1}{4} + \frac{1}{4^2} + \ldots + \frac{1}{4^n} = \frac{1 – (\frac{1}{4})^{n+1} }{ 1 – \frac{1}{4}}$$ at $n \to \infty$ gives 4/3?

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$

This question already has an answer here: Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$ 4 answers

Value of $\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}$

I remember that a couple of years ago a friend showed me and some other people the following expression: $$\lim_{n\to \infty}\frac{1^n+2^n+\cdots+(n-1)^n}{n^n}.$$ As shown below, I can prove that this limit exists by the monotone convergence theorem. I also remember that my friend gave a very dubious “proof” that the value of the limit is $\frac{1}{e-1}$. […]

Prove that the limit definition of the exponential function implies its infinite series definition.

Here’s the problem: Let $x$ be any real number. Show that $$ \lim_{m \to \infty} \left( 1 + \frac{x}{m} \right)^m = \sum_{n=0}^ \infty \frac{x^n}{n!} $$ I’m sure there are many ways of pulling this off, but there are 3 very important hints to complete the exercise in the desired manner: Expand the left side as […]

Methods to find $\lim\limits_{n\to\infty}\frac1n\sum\limits_{k=1}^nn^{1/k} $

What would you suggest here? $$\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^{n} n^{1/k} $$