An Infinite series is the sum of the values in an infinite sequence of number.
Let {aₙ} be a sequence of real numbers.
The expression a₁+a₂+a₃+a₄+.....+aₙ..... is called an infinite series and is denoted by or by ∑aₙ and aₙ is called the nth term of the series.
partial sum
we define
s₁=a₁
s₂=a₁+a₂
s₃=a₁+a₂+a₃
... ... ... ... ....
... ... ... ... ....
sₙ=a₁+a₂+a₃+....+aₙ=
There s's are called the partial Sums of the series ∑aₙ. The nth partial sum is denoted by Sₙ or sₙ.
The behaviour of the infinity series ∑aₙ is the same as that of the sequence {sₙ}. In other words, the series ∑aₙ is said to be convergent, divergent or oscillating as the sequence {sₙ} converges, diverges or oscillates.
1. Convergent series : The series ∑aₙ is said to converge if the sequence {sₙ} of its partial sums converges to s and s is called the sum of the convergent infinite series
Let {aₙ} be a sequence of real numbers.
The expression a₁+a₂+a₃+a₄+.....+aₙ..... is called an infinite series and is denoted by or by ∑aₙ and aₙ is called the nth term of the series.
partial sum
we define
s₁=a₁
s₂=a₁+a₂
s₃=a₁+a₂+a₃
... ... ... ... ....
... ... ... ... ....
sₙ=a₁+a₂+a₃+....+aₙ=
There s's are called the partial Sums of the series ∑aₙ. The nth partial sum is denoted by Sₙ or sₙ.
Behaviour of an Infinite series
The behaviour of the infinity series ∑aₙ is the same as that of the sequence {sₙ}. In other words, the series ∑aₙ is said to be convergent, divergent or oscillating as the sequence {sₙ} converges, diverges or oscillates.
1. Convergent series : The series ∑aₙ is said to converge if the sequence {sₙ} of its partial sums converges to s and s is called the sum of the convergent infinite series
here sₙ→s as n→∞
2. Divergent series : The series ∑aₙ is said to diverge to ∞ if sequence {sₙ} diverges to ∞. We write
The series ∑aₙ is said to diverge to -∞ if sequence {sₙ} diverge to -∞. We write
3. Oscillatory series : The series ∑aₙ is said to oscillate finitely or infinitely if the sequence {sₙ} oscillate finite ( or finitely ) if the sequence {sₙ} is bounded ( or unbounded ).
4. Absolute convergent series : The series ∑aₙ is said to converge absolutely if the series i.e. the series |a₁|+|a₂|+|a₃|+.....+|aₙ|+.... is convergent.
5. Conditionally convergent series : The series ∑aₙ is said to conditionally convergent if ∑aₙ is convergent but ∑|aₙ| 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 aₙ 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 {aₙ} is monotone decreasing sequence of positive terms and converges to zero, then is convergent
Three points you need to know
- aₙ>0 ∀ n
- aₙ ≥ aₙ₊₁ ∀ n and
- aₙ→0 as n→∞
3. Comparison test for series of positive terms
Test 1. Let ∑aₙ and ∑bₙ be positive terms series. Let k be a positive constant independent of n and m be a fixed positive integer.
- If aₙ ≤ kbₙ ∀ n ≥ m and ∑bₙ is convergent, then ∑aₙ is convergent.
- If aₙ ≥ kbₙ ∀ n ≥ m and ∑bₙ is divergent, then ∑aₙ is divergent.
Test 2. Let ∑aₙ and ∑bₙ be two positive terms series.
* If .l being a finite non- zero constant, then ∑aₙ and ∑bₙ both converge or diverge together.
* If and∑bₙ converges, then ∑aₙ also converges.
* If and∑bₙ diverges, then ∑aₙ also diverges.
* If and∑bₙ converges, then ∑aₙ also converges.
* If and∑bₙ diverges, then ∑aₙ also diverges.
4. P-Test
The series
converge if p>1 and diverge if p≤1
If , aₙ≥, the series is convergent if l<1 and divergent if l>1.
If , then the positive terms series ∑aₙ converges if l<1 and diverges if l>1.
If ∑aₙ be a positive term series, then ∑aₙ converges if and diverges if
If aₙ>0 and , then ∑aₙ converges for μ>1 and diverges for μ≤1.
If aₙ>0, then the series ∑aₙ
converge if p>1 and diverge if p≤1
5. Cauchy's Root Test
If , aₙ≥, the series is convergent if l<1 and divergent if l>1.
6.D′Alembert′s Ratio Test
If , then the positive terms series ∑aₙ converges if l<1 and diverges if l>1.
7. Raabe's Test
If ∑aₙ be a positive term series, then ∑aₙ converges if and diverges if
8. Gauss′s Test
If aₙ>0 and , then ∑aₙ converges for μ>1 and diverges for μ≤1.
9. Logarithmic Test
If aₙ>0, then the series ∑aₙ
Thank you for help
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