This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 2 based on Convergence of a sequence. Let's give it a try !!
Let be a sequence of real numbers such that
for some
(a) Show that
(b) Hence, or otherwise, show that converges and find the limit.
Limit
Sequence
Linear Difference Equation
(a) We are given that for some
So, ---- (1)
Again using (1) we have .
Now putting this in (1) we have , .
So, proceeding like this we have for all
and for some
---- (2)
So, from (2) we have ,
and
Adding all the above n equation we have
Hence , (proved ) .
(b) As we now have an explicit form of ----(3)
Hence from (3) we can say is bounded and monotonic ( verify ) so , it's convergent .
Now let's take both side of (3) we get ,
.
Since , as ,
.
and
are two natural number and
and
are two sets of positive real numbers such that
=
for all natural number Then prove that
and
.
This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 2 based on Convergence of a sequence. Let's give it a try !!
Let be a sequence of real numbers such that
for some
(a) Show that
(b) Hence, or otherwise, show that converges and find the limit.
Limit
Sequence
Linear Difference Equation
(a) We are given that for some
So, ---- (1)
Again using (1) we have .
Now putting this in (1) we have , .
So, proceeding like this we have for all
and for some
---- (2)
So, from (2) we have ,
and
Adding all the above n equation we have
Hence , (proved ) .
(b) As we now have an explicit form of ----(3)
Hence from (3) we can say is bounded and monotonic ( verify ) so , it's convergent .
Now let's take both side of (3) we get ,
.
Since , as ,
.
and
are two natural number and
and
are two sets of positive real numbers such that
=
for all natural number Then prove that
and
.