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Each point in $[0,1)$ has probability $p=\prod_{i=1}^\infty (1-2^{-i})$ of being missed by all the intervals, so the expected measure of what is not missed is $1-p$. This is approximately 0.7112119; …

This problem is distilled from one arising in a study of complex random variables, but I've removed as much baggage as I can without (I hope) making it trivial. Fix $D>0$. A function $f:\mathbb R\to\ …

There isn't any general method. What you usually need to do in practice is like Robert illustrated. The function is bounded on compact sets on which the function is continuous, so you need to focus on …

If you want a more precise value you can expand $k^{-s}$ around $k=N/2$ and sum term by term. I get $$\frac{2^N}{(N/2)^s}\Bigl(1 + \frac{s(1+s)}{2N} + \frac{s(1+s)(2+s)(3+s)}{8N^2} + O(N^{-3})\Bigr). …

(This question should be on math.stackexchage.com.) Substitute $v=t^{-1/2}u$, then it becomes $$ t^{-1/2} \int_0^{t^{1/2}} e^{-2u^2}\bigl(1 + O(u^3/t^{1/2})\bigr)\,du = \sqrt{\frac{\pi}{8t}} + O(t^{-1 …

Take Borel measure on $[0,1]$ as an example. Cut off disjoint intervals $I_1,I_2,\dots$ where $I_i$ has length $2^{-i}\epsilon$. That's length $\epsilon$ altogether. In the remaining $1-\epsilon$, ta …

Since the case $x=1$ is trivial, assume $x>1$ and divide both sides by $x-1$. It becomes $$(x+1)^{p-1}\ge \frac{x^p-1}{x-1} = 1 + x + x^2 + \cdots + x^{p-1}.$$ Expand the left side by the binomial the …

$\sum_{t=1}^T X_t$ is the sum of $t$ independent random variables, for example $\sum_{t=1}^4 X_t = \frac{25}{12}R_1 + \frac{13}{12}R_2 + \frac{7}{12}R_3 + \frac{1}{4}R_4$. To get a Hoeffding-type tai …

It isn't clear if you intend that $f$ and $g$ are eventually zero at the same places. Otherwise I wouldn't want to call them asymptotically equivalent. What you need is $$ f(x) = (1+o(1)) g(x), $$ w …

THIS ANSWER IS INCORRECT, SEE THE COMMENTS. According to Maple, $$ f(x) = -\frac{\gamma+\log x}{x} +\tfrac14 e^2 x - \tfrac19 e^3x^2 + \tfrac{1}{32}e^4x^3 - \tfrac{1}{150}e^5x^4 + O(x^5).$$ A converge …

The following question is a simple case of a type of problem that occurs in combinatorial enumeration problems. Define $$F(x_1,\ldots,x_n) = \frac{1}{(2\pi)^{n/2}}\exp\biggl( -\frac12\sum_{j=1}^n x_ …

For a real function $f$ defined on the reals, and real $y$, define the 2-way infinite sum $$ F_f(y) = \sum_{i\in\mathbb Z} f(y+i). $$ If $F_f(y)$ is defined for all $y$, it is periodic of period 1. …

(This is a comment, not an answer.) If $f_n(x)$ is your polynomial, starting with $f_0(x)=1$, then $$ \sum_{n=0}^\infty f_n(x) y^n = \frac{1-xy+x^2y^2+x^2y^3}{(1+xy^2)(1-xy-xy^2)} = 1 + \frac{x …

Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$. For a partition $\lambda= (j_1,\l …

More generally, if $1\le k\le N-1$ is an integer, where $N$ is a positive interger, $$S_{N,k} := \sum_{n=0}^\infty\biggl( \frac{1}{(N n+N-k)^2} + \frac{1}{(N n+k)^2} \biggr) = \frac{\pi^2}{N^2\sin^ …