INFINITE SERIES -1
Series is basically collection of terms following a particular pattern.
Infinite series are those series which infinite no. of terms.
Didn't get that ?
Let's us try to it a little bit more interesting......
Consider a thought experiment , you have decided that you will jog continuously till you lose your belly fat , but with a particular pattern . According to that pattern you will jog 1KM in first hour then 1/2 KM in second hour, 1/4KM in third hour and so on.....
Distance you travelled to lose your belly fat = 1+ 1/2 + 1/4 + 1/8 +1/16+......
Above is an example of infinite series . Total distance you travelled can be summarised mathematically by
Sn = ∑n=0(1/2n)
where n tends to infinity
Consider another thought experiment , you are on a beach and you decided that you will take out all the sand from it by using a glass which has capacity of 1000 Ltrs ,but with a particular pattern.
According to that pattern you will take 1000L in first time then 1000/2 Lin second time, 1000/3L in third time , 1000/4 L in forth time and so on.....
Volume you took out = 1000 ( 1 + 1/2 + 1/3 + 1/4 +......)
Above is another example of infinite series. Total volume of sand you took out can be summarised mathematically by
Sn = 1000∑ n=1 (1/n)
where n tends to infinity
So, we are clear with the concepts of infinite series.
Now there is a basic question that can be arisen in this case i.e.
What is the sum of such series?
The very obvious answer that comes from students' side is infinity which roughly means cannot be counted i.e. such a large no. that counted
The is half correct as it may not infinite. It depends on the type of series.
Again consider a thought experiment, you have a square and you have decided to make cuts on the square such that every succeeding piece has half of the area of preceding one.
As shown
In above if add all the cut outs after making infinite terms you will not get more than area of original square.
Here series goes like this
A + A/2 +A/4 +A/8 +A /16.........
It is geometric progression , which means succeeding and preceding term has constant ratio.
A G.P. can be defined as a progression of which terms are in a constant ratio.It could be increasing or decreasing.
Example - 3,9,27,81....so on. is an example of increasing. Here const. ratio is 3
-3,-9,-27......so on is an example of decreasing. Here const. ratio is 3
A G.P. can be given by
a , ar , ar2,ar3......
where r is the constant ratio .
Sn be the sum of n terms. There is a particular trick to get this sum. I am doing this r <1
sn = a + ar + ar2 + ar3......ar^n-1
- rsn = -ar - ar2 -........... ar^n
(1-r) Sn = a(1-rn)
Similarly it could be re-imagined by considering r> 1 keeping in mind that sum of positive must be positive.
After doing that we found following result
Series is basically collection of terms following a particular pattern.
Infinite series are those series which infinite no. of terms.
Didn't get that ?
Let's us try to it a little bit more interesting......
Consider a thought experiment , you have decided that you will jog continuously till you lose your belly fat , but with a particular pattern . According to that pattern you will jog 1KM in first hour then 1/2 KM in second hour, 1/4KM in third hour and so on.....
Distance you travelled to lose your belly fat = 1+ 1/2 + 1/4 + 1/8 +1/16+......
Above is an example of infinite series . Total distance you travelled can be summarised mathematically by
Sn = ∑n=0(1/2n)
where n tends to infinity
Consider another thought experiment , you are on a beach and you decided that you will take out all the sand from it by using a glass which has capacity of 1000 Ltrs ,but with a particular pattern.
According to that pattern you will take 1000L in first time then 1000/2 Lin second time, 1000/3L in third time , 1000/4 L in forth time and so on.....
Volume you took out = 1000 ( 1 + 1/2 + 1/3 + 1/4 +......)
Above is another example of infinite series. Total volume of sand you took out can be summarised mathematically by
Sn = 1000∑ n=1 (1/n)
where n tends to infinity
So, we are clear with the concepts of infinite series.
Now there is a basic question that can be arisen in this case i.e.
What is the sum of such series?
The very obvious answer that comes from students' side is infinity which roughly means cannot be counted i.e. such a large no. that counted
The is half correct as it may not infinite. It depends on the type of series.
Again consider a thought experiment, you have a square and you have decided to make cuts on the square such that every succeeding piece has half of the area of preceding one.
As shown
In above if add all the cut outs after making infinite terms you will not get more than area of original square.
Here series goes like this
A + A/2 +A/4 +A/8 +A /16.........
It is geometric progression , which means succeeding and preceding term has constant ratio.
A G.P. can be defined as a progression of which terms are in a constant ratio.It could be increasing or decreasing.
Example - 3,9,27,81....so on. is an example of increasing. Here const. ratio is 3
-3,-9,-27......so on is an example of decreasing. Here const. ratio is 3
A G.P. can be given by
a , ar , ar2,ar3......
where r is the constant ratio .
Sn be the sum of n terms. There is a particular trick to get this sum. I am doing this r <1
sn = a + ar + ar2 + ar3......ar^n-1
- rsn = -ar - ar2 -........... ar^n
(1-r) Sn = a(1-rn)
Similarly it could be re-imagined by considering r> 1 keeping in mind that sum of positive must be positive.
After doing that we found following result
For r >1 as n tends to infinity
Sn also tends to infinity
For r<1 as n tends to infinity
Sn = a / (1-r) rn will tends to 0
where a is the first term
From this we could state you could lose your belly fat after jogging just 2 Kms.
BE THANKFUL TO GOD THAT YOU NEEDNOT TO RUN INFINITE KMS!!!!!
Let's come back to our problem. Till Now we discovered that there are two type of series
1. Those which converges at a particular point CONVERGING SERIES
2. Those which diverges DIVERGING SERIES.
Now our problem is to determine whether the series is a converging one or diverging one.
We have some tests to determine that :
1.Comparison Test
2.Ratio Test
3.Integral Test
4.Special Comparison Test
Before doing these tests we should do PRELIMINARY TEST
In this test we convert the in terms of series as a function of 'n'. Then we check what happens when n tends to infinity.
Then if they also tends to infinity then series is a diverging one, if NOT GO FOR SOME OTHER TEST.
Example -
Series like 2, 4, 8, 16,......
'n'th term = Tn = 2n
As n tends to infinite Tn also tends to Infinity . So it is a diverging series.
Let us consider another series.
1 +1/2 + 1/3 +1/4...............
Tn = 1/n
As n tends to infinity Tn tends to 0.
This means that we need to go for other test.
REASON FOR ABOVE CONCLUSIONS
When Tn tends to infinity we said that the series is diverging one that seems very obvious . But problem arises when we said that when Tn doesn't tends to infinity why can't we conclude anything?
This is because in such series always there is addition of terms which have enough magnitude to increase total sum value....
Let's take an example:
Consider a series as following
1 + 0.0001 +0.00000001 + 0.000000000001 .......so on
Even after adding nearly 10,000
It just become equal to nearly 1.0001.......
Tn tends to 0
But In following series
1+1 +1 +1+1....... so on
After adding just 20 terms we get 20 .
Tn tends to 1.
So, we cannot use the preliminary test to test all type of series. It is limited.
1. Comparison Test
It is the most important and the basic test. It gives birth to other tests.
Generally experienced mathematicians use it to determine whether series is converging or diverging.
In this test we use known series to test its nature.
For Example - We know that you need not jog infinite kilometers to lose your belly
which imples
1 + 1/2 + 1/4 ..... is a converging series.
Any series whose terms tends to a smaller value than the above one will be converging series.
Let us elaborate this point .......
Let us take series
1 + 1/3 + 1/9 + 1/27...........
which could be summarised as
Sn = ∑n=0(1/3n)
We can easily tell that for same value of n
second series will have smaller value .
As we know that the first series converges we could easily tell that second series will also converge because for any n second series will have less value than first one.
As the formula of sum of GP states that it has to converge at 3/2 , which proves our point.
To test whether a series is diverging or not we have to check whether the given series tends to larger value than any diverging series.
Let us elaborate this point.......
For Example - We know that
1, 2, 4, 8,......... is a diverging series from the preliminary test.
Let us take a series
1,3,9,27,......
Sn = ∑n=0 3n
We could easily compare the two series and tell that as n increases second series will become larger than the first one.
So, the second one is also diverging.
It is so because the sum of first series tends to infinity and second series has to a bigger value than that.
2.Ratio Test
It is just the modification of comparison test .
It states that first find the ratio of Tn+1 and Tn.
Consider that ratio as r
if r<1 for n tending to infinity then series is converging
if r>1 for n tending to infinity then series is diverging
if r=1 for n tending to infinity then go for another test.
For Example- Let us take a series
1/1! + 1/2! +1/3! +1/4!./........
{ 1!= 1, 2! = 1*2 , 3!= 1*2*3 , 4! = 1*2*3*4......so on}
Which can be summarised as
Sn = ∑n=11/n!
Now consider a situation that a energy source generator is generating energy as above series as energy per second and we want to calculate whether the total energy is infinite or a finite value. So first we need to find whether this series is converging or diverging.
Let us do it by ratio test......
r = Tn+1 / Tn
= 1/(n+1)! ⨸ 1/n!
= n! / (n+1)!
= n(n-1)(n-2)(n-3)................ / (n+1)n(n-1)(n-2)............
= 1/n+1
As n will tend to infinity r will tend to 0
This implies energy has a finite value.
Now let us check a previous pending case of sand at the beach.
i.e. the series which has summary as
Sn = 1000∑ n=1 (1/n)
r = (1/n+1 ⨸ 1/n)
= n/n+1
= 1 / ( 1 + (1/n))
as n tends to infinity r tends to 1.
SORRY !!!! Its the third time we are failing to determine the type of series.
PROOF OF RATIO TEST
After observing the above procedure we could state that we are finding limits of ratio Tn+1 and Tn.
For those who dont know what limit actually is
Its a mathematical tool about which we will talk later in detail. But , For Now Its just a method with we were able to find the no. to which r is tending.It has exactly the same procedure that we to simplify but with a f**king type of symbol which has no meaning.
Now as we finding the limit of r an n tends to infinity let us denote it by L .
3.Integral Test
Sn also tends to infinity
For r<1 as n tends to infinity
Sn = a / (1-r) rn will tends to 0
where a is the first term
From this we could state you could lose your belly fat after jogging just 2 Kms.
BE THANKFUL TO GOD THAT YOU NEEDNOT TO RUN INFINITE KMS!!!!!
Let's come back to our problem. Till Now we discovered that there are two type of series
1. Those which converges at a particular point CONVERGING SERIES
2. Those which diverges DIVERGING SERIES.
Now our problem is to determine whether the series is a converging one or diverging one.
We have some tests to determine that :
1.Comparison Test
2.Ratio Test
3.Integral Test
4.Special Comparison Test
Before doing these tests we should do PRELIMINARY TEST
In this test we convert the in terms of series as a function of 'n'. Then we check what happens when n tends to infinity.
Then if they also tends to infinity then series is a diverging one, if NOT GO FOR SOME OTHER TEST.
Example -
Series like 2, 4, 8, 16,......
'n'th term = Tn = 2n
As n tends to infinite Tn also tends to Infinity . So it is a diverging series.
Let us consider another series.
1 +1/2 + 1/3 +1/4...............
Tn = 1/n
As n tends to infinity Tn tends to 0.
This means that we need to go for other test.
REASON FOR ABOVE CONCLUSIONS
When Tn tends to infinity we said that the series is diverging one that seems very obvious . But problem arises when we said that when Tn doesn't tends to infinity why can't we conclude anything?
This is because in such series always there is addition of terms which have enough magnitude to increase total sum value....
Let's take an example:
Consider a series as following
1 + 0.0001 +0.00000001 + 0.000000000001 .......so on
Even after adding nearly 10,000
It just become equal to nearly 1.0001.......
Tn tends to 0
But In following series
1+1 +1 +1+1....... so on
After adding just 20 terms we get 20 .
Tn tends to 1.
So, we cannot use the preliminary test to test all type of series. It is limited.
1. Comparison Test
It is the most important and the basic test. It gives birth to other tests.
Generally experienced mathematicians use it to determine whether series is converging or diverging.
In this test we use known series to test its nature.
For Example - We know that you need not jog infinite kilometers to lose your belly
which imples
1 + 1/2 + 1/4 ..... is a converging series.
Any series whose terms tends to a smaller value than the above one will be converging series.
Let us elaborate this point .......
Let us take series
1 + 1/3 + 1/9 + 1/27...........
which could be summarised as
Sn = ∑n=0(1/3n)
We can easily tell that for same value of n
second series will have smaller value .
As we know that the first series converges we could easily tell that second series will also converge because for any n second series will have less value than first one.
As the formula of sum of GP states that it has to converge at 3/2 , which proves our point.
To test whether a series is diverging or not we have to check whether the given series tends to larger value than any diverging series.
Let us elaborate this point.......
For Example - We know that
1, 2, 4, 8,......... is a diverging series from the preliminary test.
Let us take a series
1,3,9,27,......
Sn = ∑n=0 3n
We could easily compare the two series and tell that as n increases second series will become larger than the first one.
So, the second one is also diverging.
It is so because the sum of first series tends to infinity and second series has to a bigger value than that.
2.Ratio Test
It is just the modification of comparison test .
It states that first find the ratio of Tn+1 and Tn.
Consider that ratio as r
if r<1 for n tending to infinity then series is converging
if r>1 for n tending to infinity then series is diverging
if r=1 for n tending to infinity then go for another test.
For Example- Let us take a series
1/1! + 1/2! +1/3! +1/4!./........
{ 1!= 1, 2! = 1*2 , 3!= 1*2*3 , 4! = 1*2*3*4......so on}
Which can be summarised as
Sn = ∑n=11/n!
Now consider a situation that a energy source generator is generating energy as above series as energy per second and we want to calculate whether the total energy is infinite or a finite value. So first we need to find whether this series is converging or diverging.
Let us do it by ratio test......
r = Tn+1 / Tn
= 1/(n+1)! ⨸ 1/n!
= n! / (n+1)!
= n(n-1)(n-2)(n-3)................ / (n+1)n(n-1)(n-2)............
= 1/n+1
As n will tend to infinity r will tend to 0
This implies energy has a finite value.
Now let us check a previous pending case of sand at the beach.
i.e. the series which has summary as
Sn = 1000∑ n=1 (1/n)
r = (1/n+1 ⨸ 1/n)
= n/n+1
= 1 / ( 1 + (1/n))
as n tends to infinity r tends to 1.
SORRY !!!! Its the third time we are failing to determine the type of series.
PROOF OF RATIO TEST
After observing the above procedure we could state that we are finding limits of ratio Tn+1 and Tn.
For those who dont know what limit actually is
Its a mathematical tool about which we will talk later in detail. But , For Now Its just a method with we were able to find the no. to which r is tending.It has exactly the same procedure that we to simplify but with a f**king type of symbol which has no meaning.
Now as we finding the limit of r an n tends to infinity let us denote it by L .
3.Integral Test
No comments:
Post a Comment