An Infinite series is the sum of the values in an infinite sequence of number.
Let {aâ‚™} be a sequence of real numbers.
The expression a₁+a₂+a₃+a₄+.....+aâ‚™..... is called an infinite series and is denoted by or by ∑aâ‚™ and aâ‚™ is called the nth term of the series.
partial sum
we define
s₁=a₁
s₂=a₁+a₂
s₃=a₁+a₂+a₃
... ... ... ... ....
... ... ... ... ....
sâ‚™=a₁+a₂+a₃+....+aâ‚™=
There s's are called the partial Sums of the series ∑aâ‚™. The nth partial sum is denoted by Sâ‚™ or sâ‚™.
The behaviour of the infinity series ∑aâ‚™ is the same as that of the sequence {sâ‚™}. In other words, the series ∑aâ‚™ is said to be convergent, divergent or oscillating as the sequence {sâ‚™} converges, diverges or oscillates.
1. Convergent series : The series ∑aâ‚™ is said to converge if the sequence {sâ‚™} of its partial sums converges to s and s is called the sum of the convergent infinite series
Let {aâ‚™} be a sequence of real numbers.
The expression a₁+a₂+a₃+a₄+.....+aâ‚™..... is called an infinite series and is denoted by or by ∑aâ‚™ and aâ‚™ is called the nth term of the series.
partial sum
we define
s₁=a₁
s₂=a₁+a₂
s₃=a₁+a₂+a₃
... ... ... ... ....
... ... ... ... ....
sâ‚™=a₁+a₂+a₃+....+aâ‚™=
There s's are called the partial Sums of the series ∑aâ‚™. The nth partial sum is denoted by Sâ‚™ or sâ‚™.
Behaviour of an Infinite series
The behaviour of the infinity series ∑aâ‚™ is the same as that of the sequence {sâ‚™}. In other words, the series ∑aâ‚™ is said to be convergent, divergent or oscillating as the sequence {sâ‚™} converges, diverges or oscillates.
1. Convergent series : The series ∑aâ‚™ is said to converge if the sequence {sâ‚™} of its partial sums converges to s and s is called the sum of the convergent infinite series
here sâ‚™→s as n→∞
2. Divergent series : The series ∑aâ‚™ is said to diverge to ∞ if sequence {sâ‚™} diverges to ∞. We write
The series ∑aâ‚™ is said to diverge to -∞ if sequence {sâ‚™} diverge to -∞. We write
3. Oscillatory series : The series ∑aâ‚™ is said to oscillate finitely or infinitely if the sequence {sâ‚™} oscillate finite ( or finitely ) if the sequence {sâ‚™} is bounded ( or unbounded ).
4. Absolute convergent series : The series ∑aâ‚™ is said to converge absolutely if the series i.e. the series |a₁|+|a₂|+|a₃|+.....+|aâ‚™|+.... is convergent.
5. Conditionally convergent series : The series ∑aâ‚™ is said to conditionally convergent if ∑aâ‚™ is convergent but ∑|aâ‚™| is not convergent.
6. Non-convergent series : Series which diverge or oscillate are said to be non-convergent.
Nth term test
If the series is convergent, then
Important. The converse of this theorem is false. The convergence of aâ‚™ to zero does not imply that the series converges.
For example:- The harmonic series diverges, although
Equivalently, if or this limit does not exist, then the series is divergent.
Test on infinite series
1. Infinite geometric series
- converges to
- diverge to ∞ if r≥1
- oscillates finitely between 0 and 1 if r =-1
- oscillates infinitely if r<-1
- converges absolutely for | r |<1
2. Leibnitz test (alternating series)
A series whose terms are alternatively positive and negative, is called an alternative series
For examples is an alternating series
Definition:- If {aâ‚™} is monotone decreasing sequence of positive terms and converges to zero, then is convergent
Three points you need to know
- aâ‚™>0 ∀ n
- aâ‚™ ≥ aâ‚™₊₁ ∀ n and
- aâ‚™→0 as n→∞
3. Comparison test for series of positive terms
Test 1. Let ∑aâ‚™ and ∑bâ‚™ be positive terms series. Let k be a positive constant independent of n and m be a fixed positive integer.
- If aâ‚™ ≤ kbâ‚™ ∀ n ≥ m and ∑bâ‚™ is convergent, then ∑aâ‚™ is convergent.
- If aâ‚™ ≥ kbâ‚™ ∀ n ≥ m and ∑bâ‚™ is divergent, then ∑aâ‚™ is divergent.
Test 2. Let ∑aâ‚™ and ∑bâ‚™ be two positive terms series.
* If .l being a finite non- zero constant, then ∑aâ‚™ and ∑bâ‚™ both converge or diverge together.
* If and∑bâ‚™ converges, then ∑aâ‚™ also converges.
* If and∑bâ‚™ diverges, then ∑aâ‚™ also diverges.
* If and∑bâ‚™ converges, then ∑aâ‚™ also converges.
* If and∑bâ‚™ diverges, then ∑aâ‚™ also diverges.
4. P-Test
The series
converge if p>1 and diverge if p≤1
If , aâ‚™≥, the series is convergent if l<1 and divergent if l>1.
If , then the positive terms series ∑aâ‚™ converges if l<1 and diverges if l>1.
If ∑aâ‚™ be a positive term series, then ∑aâ‚™ converges if and diverges if
If aâ‚™>0 and , then ∑aâ‚™ converges for μ>1 and diverges for μ≤1.
If aâ‚™>0, then the series ∑aâ‚™
converge if p>1 and diverge if p≤1
5. Cauchy's Root Test
If , aâ‚™≥, the series is convergent if l<1 and divergent if l>1.
6.D′Alembert′s Ratio Test
If , then the positive terms series ∑aâ‚™ converges if l<1 and diverges if l>1.
7. Raabe's Test
If ∑aâ‚™ be a positive term series, then ∑aâ‚™ converges if and diverges if
8. Gauss′s Test
If aâ‚™>0 and , then ∑aâ‚™ converges for μ>1 and diverges for μ≤1.
9. Logarithmic Test
If aâ‚™>0, then the series ∑aâ‚™
Thank you for help
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