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Results tagged with limits-and-convergence
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user 172802
Convergence of series, sequences and functions and different modes of convergence.
1
vote
Accepted
Convergence rate of a sequence
Suppose that we already know that $x_k>0\forall k$. Clearly $(x_k)$ and $(y_k)$ are decreasing sequences which converge to $0$. Then we can prove that $b_k:=\frac{x_{k}}{x_{k-1}}\to 1$. To do it note …
- 10.6k
7
votes
Accepted
Convergence of a sequence
If, as you say, $a_k<0.1$ for all $k$, then we can prove by induction that $x_k>\frac{3}{4}$ for all $k$, with induction step $x_{k+1}>0.9\left(\frac{3}{2}-\frac{1}{2\cdot\frac{3}{4}}\right)=\frac{3}{ …
- 10.6k
5
votes
Convergence of sequences formed by orthocenters, incenters, and centroids in repeated triang...
Below I give an example of a triangle which seems to grow exponentially when iterating the process from the question (so it satisfies Scenario 3). The example relies on the existence of a fixed point …
- 10.6k