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An series related to the eta Dedekind function
Prove that
Solution
Let us consider the function
and integrate it along a quadratic counterclockwise contour with vertices
where
is a big odd natural number. Hence,
We note that and that
.
It’s also easy to see that
Hence, as we have that
By Residue theorem we have that
It is straightforward to show that
Hence,
in the limit . The result follows.
Convergence of series
Let denote the greatest prime factor of
. For example
,
. Define
. Examine if the sum
converges.
Solution
Let be the
-th prime number.
If , then
with
,
and
. It folows that
From Merten’s theorem
and the original series has the same character as
which is convergent.
On a prime summation
Let denote the
– th prime. Evaluate the sum
Square summable
Let be a real sequence such that
If
is a real sequence that is square summable; i.e
the sequence
converges.
Prove that is also square summable.
Solution
Let be defined as
where . We note that
Equality holds when . Hence,
. From the hypothesis, it follows that
is pointwise bounded. It follows from the Uniform boundedness principle ( Banach – Steinhaus ) that
are bounded. Hence,
is square summable.
Trigonometric series
Let . Prove that
Solution
First of all we note that
Hence,
Letting we get the requested value.