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Results tagged with limits-and-convergence
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user 12176
Convergence of series, sequences and functions and different modes of convergence.
11
votes
Accepted
When does this interesting sum diverge?
In short,
$$ \begin{cases}
\text{when }1\leq x & \text{series diverges when }y\le1\\
\text{when }\frac{1}{2}<x<1 & \text{series diverges when }y\leq\frac{x}{2x-1}\\
\text{when }0<x\leq\frac{1}{2 …
- 149k
22
votes
Accepted
On prime numbers
The sums above are Riemann sums over different partitions for the function $\arctan(x)$ between $x=0$ and $x=1$, so the limit will equal the integral which is $\frac{\pi}{4}$. Both of these converge …
- 11.4k