Tagged Questions
2
votes
0answers
45 views
convergence in metric space
Let $C[-1, 1]$ be the space of continuous functions equipped with the metric $(f, g) = \displaystyle\max_{x \in [-1, 1]} |f(x)-g(x)|$.
Consider the sequence $(f_n)$ of functions $f_n : [-1, 1] \to ...
8
votes
6answers
763 views
Pointwise Convergence of $\sum \frac{\sin(\sqrt{n}x)}{n}$
I am having trouble in proving the pointwise convergence of $$ g(x)=\sum_{n=1}^\infty \frac{\sin(\sqrt{n}x)}{n}$$ for all real numbers $x$ using elementary methods (e.g. integral test, Weierstrass ...
0
votes
1answer
40 views
Accelerated Order of Convergence
Let $m > 0$ and $ a:[0,1] \rightarrow \mathbb R$ be a function with $a(\epsilon) \rightarrow 0$.
Then $ \epsilon^m a(\epsilon) \rightarrow 0 $ for $ \epsilon \rightarrow 0$.
But is $ \epsilon^m ...
