Mathematics (M101)Sequence and Series

B.Tech 1st Year 1st Semester

Sequence

Definition: An Infinite sequence is a function f(n) whose domain is an infinite subset of whole numbers.

We usually write a sequence {an} . where f(n) =  an

The following are the example of sequences

{(.5)n}    {n/(n+1)}    {en/n!} and {(-1)n}

You'll notice that sequences are discontinuous everywhere but never the less play an important role in Mathematics as we will see when we consider Infinite Series.

1. an = {(.5)n}

2. an = n/ (n+1)

3. an = {en/n!}

4. an= {(-1) n }

As you could tell from the examples the 1st and 3d converge to 0, the second converges to 1 and the 4th diverges.

But do we mean by convergence exactly?Definition:For any  ε>0 there exists a number M such that L-  ε < an < L +  ε.whenever n> M Or as is usually stated  given any  ε>0 there exists a number M such that   |  an - L | <  ε whenever n > M.Then the sequence an is said to converge to L.

How do we compute limits?

There are 2 very important theorems

1. Given {an} if there is a continuous differentiable  function f(x) such that f(n) = an

then lim an = lim f(x)  therefore we can use our results on differentiable functions and L'Hopital's Rule.

Examples 1 and 2 fit into this category. But what about the 3d? the Gamma Function notwithstanding we need the following

2.   If  {an}  is increasing and bounded above or decreasing and bounded below it converges.

The first 3 examples make this fairly obvious.

Theorem: Suppose {an} is a sequence which is increasing and bounded above, then it

converges.

Proof: Let L be the Least Upper Bound of {an} i.e. it is an upper bound and there are

none smaller.

Let be given.

Let M be the first number such that am > L -. Such a number exists otherwise L -

would be an Upper Bound.

For all n>M an > am since an is an increasing sequence.

Further for all n an < L + because if L is an upper bound L+ is an upper bound.

Therefore we have shown given any > 0 There is a number M such that

L - < an < L+ whenever n > M i.e {an} converges --In fact it converges to its Least

Upper Bound.

Alternating Sequences:

Another important theorem

Suppose bn = (-1)n an where an >0. Then  if  lim an = 0 then lim bn = 0

If lim an  ≠ 0  then  { bn} diverges.   Even if  lim an = L   The alternating sequence  (-1)n an

diverges as the subsequence of even terms converges to L and the subsequence of odd terms converges to - L therefore the sequence diverges.

The next 2 examples show the divergent case and the 3d shows convergence of an alternating sequence.

1. (-1) n (n/(n+1))

2. (-1) n * n

3. (-1) n (.5) n

Let’s start off with some terminology and definitions. 

Given any sequence  we have the following.

1. We call the sequence increasing if  for every n.

2. We call the sequence decreasing if  for every n.

3. If  is an increasing sequence or  is a decreasing sequence we call it monotonic.

4. If there exists a number m such that  for every n we say the sequence is bounded below. The number m is sometimes called a lower bound for the sequence.

5. If there exists a number M such that  for every n we say the sequence is bounded above.  The number M is sometimes called an upper bound for the sequence.

6. If the sequence is both bounded below and bounded above we call the sequence bounded.

7. An infinite sequence is a function whose domain is the set of all positive

integers.

8. A finite sequence is a function whose domain is {1, 2, 3, … , n}, the first n

positive integers.

Examples:  Determine if the following sequences are monotonic and/or bounded.

(a)

(b)    

(c)   

Solutions:1.

This sequence is a decreasing sequence (and hence monotonic) because,

                                                           for every n.   Also, since the sequence terms will be either zero or negative this sequence is bounded above.  We can use any positive number or zero as the bound, M, however, it’s standard to choose the smallest possible bound if we can and it’s a nice number.  So, we’ll choose

 since,

 This sequence is not bounded below however since we can always get below any potential bound by taking n large enough.  Therefore, while the sequence is bounded above it is not bounded. As a side note we can also note that this sequence diverges (to  if we want to be specific).

2.

The sequence terms in this sequence alternate between 1 and -1 and so the sequence is neither an increasing sequence or a decreasing sequence. Since the sequence is neither an increasing nor decreasing sequence it is not a monotonic sequence.   The sequence is bounded however since it is bounded above by 1 and bounded below by -1.   Again, we can note that this sequence is also divergent.

3. This sequence is a decreasing sequence (and hence monotonic) since,

                                                             The terms in this sequence are all positive and so it is bounded below by zero.  Also, since the sequence is a decreasing sequence the first sequence term will be the largest and

so we can see that the sequence will also be bounded above by .  Therefore, this sequence is bounded. We can also take a quick limit and note that this sequence converges and its limit is zero.

Convergence:

A sequence {an}of real (or complex) numbers is said to converge to a real (or complex)

number c if for every > 0 there is an integer N > 0 such that if n > N then | an - c | < The number c is called the limit of the sequence {a

n}and we sometimes write an c. If a sequence {a

n}does not converge, then we say that it diverges.

Consider the sequence {1/n}. It converges to zero. The sequence {(-1)n}does not converge.

The sequence converges to zero.

Series|

Definitions: Let be a sequence. Define a new sequence , called the sequence

of partial sums for the sequence , by the following equations:

, for .

This new sequence of partial sums is called the series of s, and it is denoted by .

For each , is called the partial sum for the series . The number is called

the term of the series (as well as the term of the sequence).

For , . This means

So we often write

Sometimes the index for the sequence does not begin with the number 1. In

such cases the corresponding series is also indexed accordingly. For example, if

the sequence is , then the series is ; while if

the sequence is , then the series is .

Examples: Below are some examples that illustrate the difference between a sequence and the corresponding series and the sequence of partial sums.

1) Consider the sequence , where and are constants. (If

, then the sequence is the constant sequence of Example 1. If , then the sequence is also a constant sequence of Example 1.)

Observe that . The corresponding series is

.

Calculating the sequence of partial sums:

(This last equation is verified in class!)So the sequence of partial sums is

.

The sequence is called the Geometric Sequence and the series

is called the Geometric Series, and will be used often in this course. The termis called the first term (for obvious reasons) and the number is called the

common ration since the ratio of consecutive terms is .

2) Consider the sequence .

Observe that . The

corresponding series is . For any

integer , the following equality holds by partial fractions decomposition

(remember the technique of integration?) . We apply this

equality to compute the partial sums as follows:

So the sequence of partial sums is . Notice that the equality from

the partial fractions decomposition allows for the simplification of the partial

sums. Because of the fact that the intermediate terms add out, this series is called a telescoping series.

Definitions: Let be a sequence and let denote the corresponding series. The

series is said to converge provided the sequence of partial sums converges.

The series is said to diverge provided the sequence of partial sums diverges.

The series converges if and only if for some number S. This number S is

called the sum of the series and one writes .

Examples: The examples from above are reconsidered in the context of convergence/divergence.

1) Consider the Geometric Series

; , .

From above, the sequence of partial sums is

.

So . From

our discussions on convergent sequences, exists if and only if

(recall and ). Since for , the series converges

if and only if . In this case . It is important to note that the

Geometric Series converges if and only if the common ratio is less than one in absolute value. In this case, the sum of the Geometric Series is the first term of the series divide by the quantity one minus the common ratio.

2) Consider the telescoping series

.

From above, the sequence of partial sums is . So

and the sequence of partial

sums converges to one. So the series converges and

.

Now the question is- What properties of the terms determine whether the series

converges or diverges? This question is answered by stating a number of ‘tests’ that can be applied to the terms . The first such ‘test’ is the following:

The Divergence Test: Consider the series . If , then the series

diverges.

There are several observations one must keep in mind when applying the Divergence Test.

A. The test does not say that if , then the series converges.

B. If , then one does not have any information about the

convergence/divergence of the series. The Harmonic Series is an example of a

series with but the series diverges. The telescoping series above is an

example of a series with but the series converges.

C. The Divergence Test can only show a series diverges. It never shows a series converges.

Examples: Below are examples of how we can show various series converge or diverge using the information we have so far.

1) Does the series converge or diverge for ?

Since , the series diverges by the Divergence Test.

2) Does the series converge or diverge?

Since for all n, the series is a Geometric Series

with first term and common ratio . So the

series converges and .

3) Does the series converge or diverge?

Since ,

the series diverges by the Divergence Test.

4) Does the series converge or diverge?

Since for all n, the series

is a Geometric Series with first term and

common ratio . So the series converges and

Some important infinite series:

Geometric Series:

The series is known as geometric series with common ratio x.

A Geometric series is convergent if -1<x<1, divergent if X>= 1 and oscillatory if x<= -1.The p-series:

The series is called p series.

A p series is convergent if p>1 and divergent if p<=1.

Comparison Test:

Let and be two series of positive numbers and is convergent. Then

is convergent if

i) There exist an integer N such that un <= k vn , for all n>=N , where k is a constant, or if

ii) , l is a finite number , may be zero.

Example:  Determine if the following series converges or diverges.

                                                                SolutionTo use the limit comparison test we need to find a second series that we can determine the convergence of easily and has what we assume is the same convergence as the given series.  On top of that we will need to choose the new series in such a way as to give us an easy limit to compute for c. We’ve already guessed that this series converges and since it’s vaguely geometric let’s use

                                                                   as the second series.  We know that this series converges and there is a chance that since both series have the 3n in it the limit won’t be too bad.   Here’s the limit.

                                                           

Now, we’ll need to use L’Hospital’s Rule on the second term in order to actually evaluate this limit.

                                                         So, c is positive and finite so by the Comparison Test both series must converge since

converges.

Example:  Determine if the following series converges or diverges.

                                                            SolutionFractions involving only polynomials or polynomials under radicals will behave in the same way as the largest power of n will behave in the limit.  So, the terms in this series should behave as,

                                                           and as a series this will diverge by the p-series test.  In fact, this would make a nice choice for our second series in the limit comparison test so let’s use it.

                                            So, c is positive and finite and so both limits will diverge since

                                               diverges.

D’Alembert’s Ratio Test:  Let ∑an be a series, and assume that . Then if l<

1, the series is convergent, if l > 1, the series is divergent, while if l = 1, the test gives no information.

Example:   Let an = . Then ∑an is convergent.

Solution. We look the ratio of adjacent terms in the series (of positive terms).

= =

   

 

= =     as n .    

   

Since the ratio of adjacent terms in the series tends to a limit which is < 1, the series converges by the ratio test.

Example:  Determine if the following series is convergent or divergent.

                                                                SolutionHere’s the limit.

                             Again, the ratio test tells us nothing here.  We can however, quickly use the divergence test on this.  In fact that probably should have been our first choice on this one anyway.

                                                          By the Divergence Test this series is divergent.