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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 .

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