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
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
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
Important. The converse of this theorem is false. The convergence of aₙ to zero does not imply that the series
For example:- The harmonic series
Equivalently, if
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
Definition:- If {aₙ} is monotone decreasing sequence of positive terms and converges to zero, then
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
* If
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 <1)
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
6.D′Alembert′s Ratio Test
If
7. Raabe's Test
If ∑aₙ be a positive term series, then ∑aₙ converges if
8. Gauss′s Test
If aₙ>0 and
9. Logarithmic Test
If aₙ>0, then the series ∑aₙ
Thank you for help
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