Unit - 1

Infinite series

Q1) Define sequence.

A1)
A function f: N , where S is a non-empty set, is called sequence, for each nϵN.

The sequence is written as f (1), f (2), f (3), f (4) ………. f(n).

Any sequence f(n) can be denoted as <f(n)> or {f(n)} or (f(n)).

Suppose f(n) =

Then it can be written as -    and can be denoted as <>or {} or ()

is the n’th term of the sequence.

Q2) Define convergent and divergent sequence.

A2)

Convergent sequence- A sequence Sn is said to be convergent when it tends to a finite limit. That means the limit of a sequence Sn will be always finite in case of convergent sequence.

Divergent sequence- when a sequence tends to ±∞ then it is called divergent sequence.

Q3) What is an infinite series?

A3)

Infinite series- If  is a sequence, then is called the infinite series.

It is denoted by .

Examples of infinite series-

Q4) What are the properties of infinite series?

A4)

Properties of infinite series –

1. The convergence and divergence of an infinite series is unchanged addition or deletion of a finite number of terms from it.

2. If positive terms of convergent series change their sign, then the series will be convergent.

3. Let   converges to s, let k be a non-zero fixed number then converges to ks.

4. Let converges to ‘l’ and converges to ‘m’.

Q5) Check whether the series  is convergent or divergent. Find its value in case of convergent.

A5)

The general formula for this series is given by,

Sn =  = )

We get,

) = 3/2

Hence the series is convergent and its values is 3/2.

Q6) Check whether the following series is convergent or divergent. If convergent, find its value.

A6) n’th term of the series will be,

Q7) Prove that the following series is convergent and find its sum.

A7)

Here,

And

Hence the series is convergent and the limit is 1/2.

Q8) Test the convergence of the following series.

A8)

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.

Q9) Show that the following series is convergent.

A9)

Here we have

Suppose,

Which is finite and not zero.

By comparison test and converge or diverge together.

But,

Is convergent. So that is also convergent.

Q10) Test the convergence of the series whose n’th term is given below-

n’th term =

A10)

We have

  and

By D’Alembert ratio test,

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

Q11) Test the series by integral test-

A11)

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.

Q12) Test the convergence of the following series.

Sol. Neglecting the first term the series can be written as,

are in A.P nth term

are in A.P nth term

are in A.P nth term

So that,

By ratio test  converges if |x|<1 and diverges if |x|>1, but if |x| = 1 the true. test fails,

Then,

and

By Raabe’s test  converges hence the given series is convergent when |x|≤ 1 and divergent If |x| >1.

Q13) 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.

Q14) Test the convergence of the series whose nth term is given below-

A14)

By root test is convergent.

Q15) Show that the following series is convergent.

A15)
 

By root test is convergent.

Q16) Test the convergence of the following alternating series:

A16)

Here in the series, we have

First condition-

So that,

| > ||

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

Now second condition-

Both conditions are not satisfied for convergence.

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

Q17) Show that the series is absolutely convergent.

A17)

We have,

|| =     and      || =

The first condition and second conditions are-

1.  ||<||

2.

Both the conditions are satisfied.

So that we can say that by Leibnitz’s rule, the series is convergent.

The series is also convergent by p-test as p = 2 > 1.

Hence the given series is absolutely convergent.

Q18) Show that the series is absolutely convergent.

A18)

We have,

|| =     and      || =

The first condition and second conditions are-

1.  ||< ||

2.

Both the conditions are satisfied.

So that we can say that by Leibnitz’s rule, the series is convergent.

The series is also convergent by p-test as p = 2 > 1.

Hence the given series is absolutely convergent.

Q19) Test the series for absolute/conditional convergence.

A19)

Here,

1.

2.

And,

Hence bt Leibnitz’s test, the given series is convergent,

But,

Is divergent by p-series test.

So that, the given series is conditionally convergent.

Q20) Express the polynomial in powers of (x-2).

A20)

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-

Q21) By using Maclaurin’s series expand tan x.

A21)

Let-

Put these values in Maclaurin’s series we get-

Q22) Find the asymptotes of the following curve-

A22)

Here the highest power of x is and the coefficient is

We can find the asymptote parallel to x-axis as-

And the highest power of y is and its coefficient is .

The asymptotes parallel to y-axis, are given by-

Therefore, the asymptotes are x = 0, x = 1, y = 0 and y = 1.

Q23) Find the vertical asymptote of .

A23)

has vertical asymptotes when ,

Thus x = 4 and x = -1 are the vertical asymptotes of

Q24) Find the horizontal asymptote of the function given below-

A24)

In order to find the horizontal asymptotes, we find-

Hence the horizontal asymptote is the line y = 2.

Example: Find the asymptotes parallel to x-axis and y-axis of the following curve-

Sol.

We can write the given equation as-

Q25) Find the slant asymptote of

A25)

Here we will find m and c,

Hence the slant asymptote is y = x