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# ISI MStat PSB 2009 Problem 2 | Linear Difference Equation 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 !!

## Problem- ISI MStat PSB 2009 Problem 2

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.

### Prerequisites

Limit

Sequence

Linear Difference Equation

## Solution :

(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 , .

## Food For Thought 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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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 !!

## Problem- ISI MStat PSB 2009 Problem 2

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.

### Prerequisites

Limit

Sequence

Linear Difference Equation

## Solution :

(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 , .

## Food For Thought and are two natural number and and are two sets of positive real numbers such that = for all natural number Then prove that and .

## Subscribe to Cheenta at Youtube

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