MHB How can we prove the convergence of recursive defined sequences?

mathmari
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Hey! :giggle:

a) Check the convergence of the sequence $a_n=\left (\frac{n+2000}{n-2000}\right)^n$, $n>1$. If it converges calculate the limit.
b) Check the convergence of the recursive defined sequence $a_n=\frac{a_{n-1}}{a_{n-1}+2}$, $n>1$, with $a_1=1$.For a) we have $$a_n=\left (1+\frac{4000}{n-2000}\right) ^{n-2000}\left (1+\frac{4000}{n-2000}\right) ^{2000}\to e^{4000}$$ Having found the limit means that the sequence is also convergent, right? but could we have shown the convergence also in an other way?For b) when we calculate some terms we see the sequence is decreasing and we can prove that using induction. It also holds that $a_n>0$. This means that the sequence converges, right?

:unsure:
 
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mathmari said:
a) Check the convergence of the sequence $a_n=\left (\frac{n+2000}{n-2000}\right)^n$, $n>1$. If it converges calculate the limit.

For a) we have $$a_n=\left (1+\frac{4000}{n-2000}\right) ^{n-2000}\left (1+\frac{4000}{n-2000}\right) ^{2000}\to e^{4000}$$ Having found the limit means that the sequence is also convergent, right? but could we have shown the convergence also in an other way?

Hey mathmari!

That works. (Nod)

We may want to mention which propositions we're using to conclude it though.
It's easy to make mistakes if we take the limits of parts of an expression after all. 🧐

I wouldn't immediately know a different way to do it, other than making sure the steps are correct.

mathmari said:
b) Check the convergence of the recursive defined sequence $a_n=\frac{a_{n-1}}{a_{n-1}+2}$, $n>1$, with $a_1=1$.
For b) when we calculate some terms we see the sequence is decreasing and we can prove that using induction. It also holds that $a_n>0$. This means that the sequence converges, right?
Yep. It follows from the monotone convergence theorem. (Nod)
 
Klaas van Aarsen said:
That works. (Nod)

We may want to mention which propositions we're using to conclude it though.
It's easy to make mistakes if we take the limits of parts of an expression after all. 🧐

I wouldn't immediately know a different way to do it, other than making sure the steps are correct.Yep. It follows from the monotone convergence theorem. (Nod)

Great! Thank you! (Sun)
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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