Skip to main content

All Questions

4 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
6 votes
0 answers
283 views

Is the arithmetic-geometric mean of 1 and 2 rational?

It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
Rick Does Math's user avatar
6 votes
0 answers
267 views

Convergence of $\sum_{n=1}^\infty x_n^k$

I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
Ma Joad's user avatar
  • 1,755
5 votes
0 answers
285 views

How do you go about making ranges (for integer variables) independent?

Basic question: say you have a sum $$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$ where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
H A Helfgott's user avatar
  • 20.2k
1 vote
0 answers
51 views

Mean value of a function with binomial coefficients as weights

Is the following true? Let $a$ be a positive integer and let $t_n$ be a sequence of numbers. We define the binomial mean of $t_n$ $$ \beta_{t_n,a} = \frac{1}{2^n t_n}\sum_{r^a \le n} \binom{n}{r^a}...
Nilotpal Kanti Sinha's user avatar