1
vote
1answer
40 views

is this or (when) does this equality hold for weighted power series

$s_n=\sum _{k=0}^n a_k$ for every $n$ and $x\in(0,1)$. Let $p=(p_n)$ is a sequence of nonnegative numbers with $p_0>0$ s.t $P_n= \sum_{k=0}^{n}p_k \rightarrow \infty$ as $n \rightarrow \infty$ ...
28
votes
2answers
2k views

What's the mean of all real numbers?

At first, I had thought the average must be zero, since for every positive number there's an equal magnitude negative number to cancel out the positive number's effect on the average, leaving only ...
4
votes
2answers
147 views

Basic question about natural density

Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between ...
5
votes
1answer
490 views

Harmonic mean and logarithmic mean

The harmonic mean of a finite set of positive real numbers $\{x_1, x_2, \ldots, x_n\}$ is defined to be $$H(\{x_1, x_2, \ldots, x_n\}) = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + ...
7
votes
3answers
266 views

Asymptotic difference between a function and its binomial average

The origin of this question is the identity $$\sum_{k=0}^n \binom{n}{k} H_k = 2^n \left(H_n - \sum_{k=1}^n \frac{1}{k 2^k}\right),$$ where $H_n$ is the $n$th harmonic number. Dividing by $2^n$, we ...