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Results tagged with limits-and-convergence
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user 14830
Convergence of series, sequences and functions and different modes of convergence.
18
votes
Accepted
Asymptotics of a Bernoulli-number-like function
[Revised and expanded to give the answer for all $k>1$ and incorporate
further terms of an asymptotic expansion as $n \rightarrow \infty$]
Fix $k>1$, and write $a_1=f(1,k)=1$ and
$$
a_n = f(n,k) =
\f …
- 79.9k
12
votes
Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win w...
The probability of not winning is
$$
\prod_{T=1}^\infty \left(1 - \frac1{2^T} \right)
= \frac12 \frac34 \frac78 \frac{15}{16} \cdots
= 0.28878809508660 \ldots ;
$$
that's a well-known constant (equal …
- 79.9k
8
votes
Asymptotic behavior of a certain trigonometric partial sum
The desired inequality should be true iff
$$
c < c_0 := (r - \sqrt{r^2-1})^2
\quad\ \text{where} \quad\
r = \frac{|a|}{2b}
$$
(NB the hypotheses $b>0$ and $a < -2b$ imply $r>1$, so $0 < c_0 < 1$).
Num …
- 79.9k
6
votes
On the continuity of $\sum_{n=1}^{\infty} \sin(nx) / n^\alpha$
For small $x>0$ we can write
$$
f(x) = x^{\alpha-1} \cdot x \sum_{n=1}^\infty \frac{\sin nx}{(nx)^\alpha},
$$
which is $x^{\alpha-1}$ times a Riemann sum for
$$
I_\alpha := \int_0^\infty \sin u \frac{ …
- 79.9k