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Explore the Back-StoryA sequence of real numbers is an one to one mapping from (the set of natural numbers) to a subset of (the set of all real numbers).

Sum of the terms of a real sequence sequence is called a series.

Find the sum of the series .

IIT JAM 2016, PROBLEM 24

Sequence and Series of real numbers.

6 out of 10

Introduction to Real Analysis : Robert G. Bartle & Donald R. Sherbert.

This problem is a very basic one, this problem can easily be solved by step by step solution. The steps are:

**Step 1 :** First we will ignore the summation part. We will factorize the denominator, because we are going step by step so our aim is to simplify the given problem first.

**Step 2:** After factorizing the the denominator we will reach to a position where we have to use partial fraction to go forward.

**Step 3:** In this step we will take care of the part, like how it will affect the series.

**Step 4:** After taking care of the we will now expand the summation (breaking it into infinite sum).

**Step 5 :** So after steps we are halfway done now just the last simplification is left we will use the value

to simplify it further.

Now try to solve the entire problem by following these steps !!!

Let us see how to execute **STEP 1 .**

, here we are only concerned with the denominator part so,

So we have successfully factorized the denominator. Now you can reach to the part where we have to use the concept of partial fraction!!!

Let us execute **STEP 2 **now.

After factorizing the denominator in **HINT 1** we get

Now our aim here is to seperate and so we can do this by using partial fraction.

**Now taking we get**

Again taking , we get:

therefore,

So we get,

So our aim here is successful we have separated and . Now can you proceed further ???

Now after using partial fraction we get.

Now we will execute **STEP 4 **i.e., we will take care of the and at the same time we will execute **STEP 4 **i.e, we will split it into the infinite sum.

Now after this can you applying the formula of '' to finish the problem!!

Now only **STEP 5 **is left to execute. So we will use the infinite series of .

**Hence the anser is C.**

- https://www.cheenta.com/eigen-values-of-a-matrix-iit-jam-2016-problem-number-13/
- https://www.youtube.com/watch?v=P4ZYA4XCQoM

A sequence of real numbers is an one to one mapping from (the set of natural numbers) to a subset of (the set of all real numbers).

Sum of the terms of a real sequence sequence is called a series.

Find the sum of the series .

IIT JAM 2016, PROBLEM 24

Sequence and Series of real numbers.

6 out of 10

Introduction to Real Analysis : Robert G. Bartle & Donald R. Sherbert.

This problem is a very basic one, this problem can easily be solved by step by step solution. The steps are:

**Step 1 :** First we will ignore the summation part. We will factorize the denominator, because we are going step by step so our aim is to simplify the given problem first.

**Step 2:** After factorizing the the denominator we will reach to a position where we have to use partial fraction to go forward.

**Step 3:** In this step we will take care of the part, like how it will affect the series.

**Step 4:** After taking care of the we will now expand the summation (breaking it into infinite sum).

**Step 5 :** So after steps we are halfway done now just the last simplification is left we will use the value

to simplify it further.

Now try to solve the entire problem by following these steps !!!

Let us see how to execute **STEP 1 .**

, here we are only concerned with the denominator part so,

So we have successfully factorized the denominator. Now you can reach to the part where we have to use the concept of partial fraction!!!

Let us execute **STEP 2 **now.

After factorizing the denominator in **HINT 1** we get

Now our aim here is to seperate and so we can do this by using partial fraction.

**Now taking we get**

Again taking , we get:

therefore,

So we get,

So our aim here is successful we have separated and . Now can you proceed further ???

Now after using partial fraction we get.

Now we will execute **STEP 4 **i.e., we will take care of the and at the same time we will execute **STEP 4 **i.e, we will split it into the infinite sum.

Now after this can you applying the formula of '' to finish the problem!!

Now only **STEP 5 **is left to execute. So we will use the infinite series of .

**Hence the anser is C.**

- https://www.cheenta.com/eigen-values-of-a-matrix-iit-jam-2016-problem-number-13/
- https://www.youtube.com/watch?v=P4ZYA4XCQoM

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