Question Bank

Question-1: If 

                      = =  = ½

Hence the limit of the sequence is 1/2  .

Question-2: check whether the series is convergent or divergent. Find its value in case of convergent.

Sol.  As we know that,

           Sn = 

Therefore,

         Sn =   

Now find out the limit of the sequence,

                                                        = ∞

Here the value of the limit is infinity, so that the series is divergent as sequence diverges.

Question-3: check whether the series  is convergent or divergent.         

Sol. The general formula can be written as,

We get on applying limits,

                                   ) = 3/4

This is the convergent series and its value is 3 / 4

Question-4: Test the convergence of the following series.

Sol. We have  

First we will find  and the

And

Here, we can see that, the limit is finite and not zero,

Therefore, and converges or diverges together.

Since is of the form  where p = 2>1

So that , we can say that,

is convergent , so that will also be convergent.

Question-5: Test the convergence of the following series-

Sol. Here we have the series,

Now,

Now comapare 

We can see that the limit is finite and not zero.

Here  and converges or diverges together since ,

is the form of here p = 1,

So that,

  is divergent then is also divergent.

Question-6: Show that the following series is convergent.

Sol. 

Suppose,

Which is finite and not zero.

By comparison test and converge or diverge together.

But,

Is convergent. So that is also convergent.

Question-7: Test the convergence of the following series:

Sol.

Here we take,

Which not zero and finite ,

So by comparison test , and both converges or diverges, but by p-series test

Is convergent . So that  is convergent.

Question-8: Test the convergence of the following series:

Sol.

Here we take,

Which not zero and finite ,

So by comparison test , and both converges or diverges, but by p-series test

Is convergent . So that  is convergent.

Question-9: Test for the convergence of the n’th term of the series given below-

Sol. We have ,

Now , by D’Almbert ratio test converges if and diverges if 

At x = 1 , this test fails.

Now , when x = 1

The limit is finite and not zero.

Then by comparison test, converges or diverges together.

Since is the form of , in which

Hence diverges then will also diverge.

Therefore in the given series converges if x<1 and diverges if x≥1.

Question-10: Test the convergence of the series,

Sol. As we will neglect the first term, we get

By ration test  is convergent when x<1 and divergent when x>1, when x= 1,

The rario test fails, then

By Rabee’s test is convergent , hence the given series is convergent when x≤ 1 and divergent If x >1.

Question-11: Test the nature of the following series:

Sol.

By ration test  is convergent when (x/4)<1 and divergent when x>4, when x= 4,

The rario test fails, then

By Rabee’s test is convergent , hence the given series is convergent when x<4 and divergent If x >=4.

Question-12: Test the series by integral test-

Sol. Here is positive and decreases when we increase n ,

      Now apply integral test,

Let,          

X = 1 , t = 5 and x = ∞ , t = ∞,

Now,

So by integral test,

The series is divergent.

Question-13: Test the convergence of the following series:

Sol. Here, we have,

Therefore the given series is convergent.

Question-14: Test the convergence of the following series:

Sol.  We have the series,

Here , And

Which gives,

  , the series is convergent.

If   , the series is divergent.

.

Thus the series is divergent.

Question-15: Test the convergence of the following alternating series:

Sol.  Here in the series, we have

First condition-

So that,

                            || > ||

That means , each term is not numerically less than its preceeding terms.

Now second condition-

Both conditions are not satisfied for convergence.

Hence the given series is not convergent. It is oscillatory.

Question-16: Test the following series for the convergence-

Sol. We have the given series, now

We see that, this is an alternating series,

Here,

Also,

By Leibinitz’s test the series is convergent.

Question-17: Test the convergence/Divergence of the series:

Sol. Here the given series is alternately negative and positive , which is also a geometric infinite series.

1. Suppose,

                        S = 

According to the conditions of geometric series,

Here , a = 5 , and common ratio (r) = -2/3 

Thus, we know that,

So ,                        

Sum of the series is finite , which is 3.

So we can say that the given series is convergent.

Now.

Again sum of the positive terms,

The series is geometric, then

A = 5 and  r = 2/3 , then

Sum of the series,

Sum of the series is finite then the series is convergent.

Both conditions are satisfied , then the given series is absolutely convergent

                                         Question Bank

                                         Question Bank

Question-1: If 

                      = =  = ½

Hence the limit of the sequence is 1/2  .

Question-2: check whether the series is convergent or divergent. Find its value in case of convergent.

Sol.  As we know that,

           Sn = 

Therefore,

         Sn =   

Now find out the limit of the sequence,

                                                        = ∞

Here the value of the limit is infinity, so that the series is divergent as sequence diverges.

Question-3: check whether the series  is convergent or divergent.         

Sol. The general formula can be written as,

We get on applying limits,

                                   ) = 3/4

This is the convergent series and its value is 3 / 4

Question-4: Test the convergence of the following series.

Sol. We have  

First we will find  and the

And

Here, we can see that, the limit is finite and not zero,

Therefore, and converges or diverges together.

Since is of the form  where p = 2>1

So that , we can say that,

is convergent , so that will also be convergent.

Question-5: Test the convergence of the following series-

Sol. Here we have the series,

Now,

Now comapare 

We can see that the limit is finite and not zero.

Here  and converges or diverges together since ,

is the form of here p = 1,

So that,

  is divergent then is also divergent.

Question-6: Show that the following series is convergent.

Sol. 

Suppose,

Which is finite and not zero.

By comparison test and converge or diverge together.

But,

Is convergent. So that is also convergent.

Question-7: Test the convergence of the following series:

Sol.

Here we take,

Which not zero and finite ,

So by comparison test , and both converges or diverges, but by p-series test

Is convergent . So that  is convergent.

Question-8: Test the convergence of the following series:

Sol.

Here we take,

Which not zero and finite ,

So by comparison test , and both converges or diverges, but by p-series test

Is convergent . So that  is convergent.

Question-9: Test for the convergence of the n’th term of the series given below-

Sol. We have ,

Now , by D’Almbert ratio test converges if and diverges if 

At x = 1 , this test fails.

Now , when x = 1

The limit is finite and not zero.

Then by comparison test, converges or diverges together.

Since is the form of , in which

Hence diverges then will also diverge.

Therefore in the given series converges if x<1 and diverges if x≥1.

Question-10: Test the convergence of the series,

Sol. As we will neglect the first term, we get

By ration test  is convergent when x<1 and divergent when x>1, when x= 1,

The rario test fails, then

By Rabee’s test is convergent , hence the given series is convergent when x≤ 1 and divergent If x >1.

Question-11: Test the nature of the following series:

Sol.

By ration test  is convergent when (x/4)<1 and divergent when x>4, when x= 4,

The rario test fails, then

By Rabee’s test is convergent , hence the given series is convergent when x<4 and divergent If x >=4.

Question-12: Test the series by integral test-

Sol. Here is positive and decreases when we increase n ,

      Now apply integral test,

Let,          

X = 1 , t = 5 and x = ∞ , t = ∞,

Now,

So by integral test,

The series is divergent.

Question-13: Test the convergence of the following series:

Sol. Here, we have,

Therefore the given series is convergent.

Question-14: Test the convergence of the following series:

Sol.  We have the series,

Here , And

Which gives,

  , the series is convergent.

If   , the series is divergent.

.

Thus the series is divergent.

Question-15: Test the convergence of the following alternating series:

Sol.  Here in the series, we have

First condition-

So that,

                            || > ||

That means , each term is not numerically less than its preceeding terms.

Now second condition-

Both conditions are not satisfied for convergence.

Hence the given series is not convergent. It is oscillatory.

Question-16: Test the following series for the convergence-

Sol. We have the given series, now

We see that, this is an alternating series,

Here,

Also,

By Leibinitz’s test the series is convergent.

Question-17: Test the convergence/Divergence of the series:

Sol. Here the given series is alternately negative and positive , which is also a geometric infinite series.

1. Suppose,

                        S = 

According to the conditions of geometric series,

Here , a = 5 , and common ratio (r) = -2/3 

Thus, we know that,

So ,                        

Sum of the series is finite , which is 3.

So we can say that the given series is convergent.

Now.

Again sum of the positive terms,

The series is geometric, then

A = 5 and  r = 2/3 , then

Sum of the series,

Sum of the series is finite then the series is convergent.

Both conditions are satisfied , then the given series is absolutely convergent