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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.
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
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 …
- 11.4k