Mathematics Dictionary
Dr. K. G. Shih
Infinite Sequence and Series
Subjects
Symbol Defintion
Example : Sqr(x) is square root of x
AN 18 01 |
- Infinite sequences of constants
AN 18 02 |
- Infinite series of constants
AN 18 03 |
- Prove that 0 + 0 + 0 + 0 + .... = 1
AN 18 04 |
- Harmonic series S(n) = Sum[1/n]
AN 18 05 |
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AN 18 06 |
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AN 18 07 |
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AN 18 08 |
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AN 18 09 |
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AN 18 10 |
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Answers
AN 18 01. Infinte sequences of constants
Defintion
A number of T(n) with each positive number n produces numbers
Example : T(n) = n^2
n = 1 and T(1) = 1
n = 2 and T(2) = 4
n = 3 and T(3) = 9
Hence 1,4,9, ..... are sequence
Example 1 : Write out the first 6 terms of the sequence T(n) = n/(n+1)
n = 1 and T(1) = 1/2
n = 2 and T(2) = 2/3
n = 3 and T(3) = 3/4
n = 4 and T(4) = 4/5
n = 5 and T(5) = 5/6
n = 6 and T(6) = 6/7
Example 2 : Write out the first 6 terms of the sequence T(n) = (-1)^(n+1)
n = 1 and T(1) = (-1)^2 = +1
n = 2 and T(2) = (-2)^3 = -1
n = 3 and T(3) = (-1)^4 = +1
n = 4 and T(4) = (-1)^5 = -1
n = 5 and T(5) = (-1)^6 = +1
n = 6 and T(6) = (-1)^7 = -1
Example 3 : Write out the first 6 terms of the sequence T(n) = nth prime number
n = 1 and T(1) = 2
n = 2 and T(2) = 3
n = 3 and T(3) = 5
n = 4 and T(4) = 7
n = 5 and T(5) = 11
n = 6 and T(6) = 13
Go to Begin
AN 18 02. Infinite series of constants
Defintion
The summation of an infinite squence of constants
Example : S(1) = Sum[(n^2]
n = 1 and S(1) = 1
n = 2 and S(2) = 1 + 4 = 5
n = 3 and S(3) = 1 + 4 + 9 = 14
Example 1 : Write out the sum of the series S(n) = Sum[n/(n+1)] for n = 1 to 6
n = 1 and S(1) = 1/2
n = 2 and S(2) = 1/2 + 2/3
n = 3 and T(3) = 1/2 + 2/3 + 3/4
n = 4 and T(4) = 1/2 + 2/3 + 3/4 + 4/5
n = 5 and T(5) = 1/2 + 2/3 + 3/4 + 4/5 + 5/6
n = 6 and T(6) = 1/2 + 2/3 + 3/4 + 4/5 + 5/6 + 6/7
Example 2 : Write out the sum of the series S(n) = Sum[(-1)^(n+1)] for n = 1 to 6
n = 1 and S(1) = (-1)^2 = 1
n = 2 and S(2) = (-1)^2 + (-2)^3 = 1 - 1 = 0
n = 3 and S(3) = (-1)^2 + (-2)^3 + (-1)^4 = 1 - 1 + 1 = 1
n = 4 and S(4) = (-1)^5 = 1 - 1 + 1 - 1 = 0
n = 5 and S(5) = (-1)^6 = 1 - 1 + 1 - 1 + 1 = 1
n = 6 and S(6) = (-1)^7 = 1 - 1 + 1 - 1 + 1 - 1 = 0
Example 3 : Write out the sum of the series S(n) = Sum[nth prime number] for n = 1 to 6
n = 1 and S(1) = 2
n = 2 and S(2) = 2 + 3
n = 3 and S(3) = 2 + 3 + 5
n = 4 and S(4) = 2 + 3 + 5 + 7
n = 5 and S(5) = 2 + 3 + 5 + 7 + 11
n = 6 and S(6) = 2 + 3 + 5 + 7 + 11 + 13
Go to Begin
AN 18 03. 0 + 0 + 0 + 0 + 0 + ..... = 1
Proof
0 + 0 + 0 + 0 + 0 + .....
= (1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + .....
= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + .....
= 1 + 0 + 0 + 0 + ......
= 1
Question
What is wrong in the calculation ?
For summation up to odd number of terms is true
For summation to infinite terms is uncertain and is impossible
Hence the series is diverge
Guido Ubaldus thought
This is the proof of the existance of God
Because something has been created out of nothing
Go to Begin
AN 18 04. Harmonic series S(n) = Sum[1/n]
Find the sum of the first 20 terms of the harmonic series
We can find the answer by adding the calculation of each term
S(20) = 1 + 1/2 + 1/3 + 1/4 + .... + 1/20 = 3.5977
Prove that Sum[1/n] is not convergent
S(1) = 1
S(2) = 1 + 1/2 = 3/2
S(3) = 1 + 1/2 + 1/3 = 4/2
S(4) = 1 + 1/2 + 1/3 + 1/4 = 5/2
S(5) = 1 + 1/2 + 1/3 + 1/4 + 1/5 = 5/2
S(6) = 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 = 6/2
The procedures can continue indefinitely
Hence the series is divergent
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AN 18 05.
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AN 18 06. New
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AN 18 07. Answer
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AN 18 08. Answer
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AN 18 09. Answer
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AN 18 10. Answer
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