2
votes
1answer
54 views

If a series converges, does it converge with additional log term multiplied?

If $\sum_{n} |a_n| < \infty$, is it true that $\sum_{n} |a_n\log(a_n)| < \infty$ if $0 \leq a_n \leq 1$? I am trying to see if $A$ is trace class operator, then $A \log(A)$ is also trace class ...
4
votes
1answer
94 views

Finding a specific sequence of digits in pi

Looking at the pifs project on GitHub and this question on SO has made me curious as to how feasible it is to find a specific sequence of digits within Pi. Essentially, on average, how many digits of ...
1
vote
0answers
31 views

What is the “true” entropy of a binary string?

Consider an infinite binary string $\sigma$ and define its entropy $$H_1 = -(p_0 \log_2 p_0 + p_1 \log_2 p_1)$$ with $p_i = \lim_{N\rightarrow \infty} N(i)/N$, $N(i)$ the number of $i$'s among the ...
-1
votes
3answers
145 views

probabilistically what can we say about the next throw of a coin after n throws

this may sound easy or hard or whatever but i cant seem to find anything after searching around for a similar question/answer The question is this: What can we say (probabilistically) about the next ...
7
votes
2answers
274 views

Can the entropy of a random variable with countably many outcomes be infinite?

Consider a random variable $X$ taking values over $\mathbb{N}$. Let $\mathbb{P}(X = i) = p_i$ for $i \in \mathbb{N}$. The entropy of $X$ is defined by $$H(X) = \sum_i -p_i \log p_i.$$ Is it possible ...
3
votes
1answer
250 views

Entropy of $X =\{1,2,\ldots,\infty\}$ with the probability of $\{1/2^1,1/2^2,\ldots,1/2^\infty\}$?

I'm studing for an information theory exam, maybe some of you can help me here with an exercise. What's the entropy of $X$ as $\{1,2,\ldots,n\}$ ($n$=infinity) where the probabilities are $P \{1/2^1, ...