Home » Posts tagged 'sequences'
Tag Archives: sequences
Let be a sequence such that and
Evaluate the sum
Let us consider the sequence
and observe that
This in return means,
which is the desired result. Therefore,
- The recursive relation has a closed form if-f or . In our case it is:
- The number is known as the Grafting constant.
- Under the same assumptions it holds that
Let be a function such that and is differentiable at . Let us set
Evaluate the limit .
Since is differentiable at , there is some such that
and is of course continuous.
Let . There exists such that which in return means that . Hence , for larger than it holds that
On the other hand , the sum is a Riemann sum and converges to .
Let be a real sequence such that , , and . Prove that
Lemma: Let be a bounded sequence of, say, complex numbers and let be another complex number.
where the latter means “convergence on a set of density ” i.e. there exists a subsequence such that and is dense i.e.
Proof: The proof is omitted because it is too technical.
All we need is that the logarithm is continuous and that as well as are bounded sequences.
Note: We can simplify the conditions to and .
The exercise along with the solution may be found on AoPS.com . The proof of the claim may also be found there.
Let be a sequence of positive real numbers such that
Prove that converges.
Fix and let . Then, there exist such that where . Thus,
Letting it follows that
Since this holds forall it follows that and the result follows.
Consider the sequence defined recursively as
Prove that .
Lemma: If is a sequence for which then
Proof: In Stolz theorem we set and .
It is easy to see that is is monotonically decreasing to zero. Moreover, an application of L’Hospital’s rule gives
Now, due to the lemma we have and the result follows.
Remark : The asymptotic now follows to be .
Problem: Find what inequality should satisfy such that the series