Question Bank

UNIT–3

Sequence and series

Question-1: If    , then the limit of will be,

                      = =  = 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 following series is convergent or divergent. If convergent , find its value.

Sol. n’th term of the series will be,

]  = ½

Question-4: Test the convergence of the series whose nth term is given below-

Sol.

By root test is convergent.

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

Sol. Here, we have,

Therefore the given series is convergent.

Question-6: Test the convergence of the series whose n’th term is given below-

                                      n’th term =

Sol. We have    and

By D’Alembert ratio test,

So that by D’Alembert ratio test , the series will be convergent.

Question-7: 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-8: Test the convergence of the given series:

Sol.  Here,

Take,

is divergent by p-series test. (p = 0 <1)

Question-9: 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.

Questtion-10: 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-11: If the series converges, then find the value of x.

Sol. Here

Then,

By D’Almbert’s ratio test the series is convergent for |x|<1 and divergent if |x|>1.

So at x = 1

The series becomes-

At x = -1

This is an alternately convergent series.

This is also convergent series, p = 2

Here, the interval of convergence is

Question-12: Express the polynomial in powers of (x-2).

Sol.  Here we have,

                                            f(x) =

Differentiating the function w.r.t.x-

f’(x) =

f’’(x) = 12x + 14

f’’’(x) = 12

f’’’’(x)=0

Now using Taylor’s theorem-

+ …….     (1)

Here we have, a = 2,

Put x = 2 in the derivatives of f(x), we get-

f(2) = 

f’(2) =

f’’(2) = 12(2)+14 = 38

f’’’(2) = 12   and f’’’’(2) = 0

Now put a = 2 and substitute the above values in equation(1), we get-

Question-13: Find the Taylor’s expansion of about (1 , 1) up to second degree term.

Sol. We have,

At (1 , 1)

Now by using Taylor’s theorem-

……

Suppose 1 + h = x  then h =  x – 1

                 1 + k = y   then k = y - 1

……

   =

……..

Question-14: Find the value of 5, correct to five decimal places by using the power series for .

Sol. As we know that the exponential series is-

Here we get-

Now

Question-15: Simplify

Sol. We know that-

                 So that