examine the convergence of the series 1/1.3 +2/3.5 + 3/5.7 + .....
Answers
Given : 1/1.3 +2/3.5 + 3/5.7 + .
To Find : examine the convergence of the series
Solution:
1/1.3 + 2/3.5 + 3/5.7 + ..
= n/(2n-1)(2n + 1)
n/(2n-1)(2n + 1) = A/(2n -1) + B/(2n + 1)
=> n = A(2n +1) + B(2n - 1)
=> 1/2 = A(2) = > A= 1/4
-1/2 = B(-2) => B = 1/4
1/4 ( 1/(2n + 1) + 1/(2n - 1) )
Ratio Test
L = Lim n→ ∞ | aₙ₊₁ /aₙ|
= Lim n→ ∞ {1/(2n + 3) + 1/(2n + 1) } /( 1/(2n + 1) + 1/(2n - 1) )
= Lim n→ ∞ (4n + 4)(2n + 1)(2n - 1)/(2n + 3) (2n + 1)4n
= Lim n→ ∞ (4n + 4) (2n - 1)/(2n + 3)4n
= Lim n→ ∞ ( 4 + 4/n) (2 - 1/n)/(2 + 3/n) 4
= ( 4 + 0)(2 - 0)/(2 + 0) 4
= 4 (2)/(2)(4)
= 1
if L < 1 then convergent , if L > 1 then divergent
as L = 1 no conclusion can be made about convergence or divergence
Lets check
Lim n→ ∞ aₙ
Lim n→ ∞ 1/4 ( 1/(2n + 1) + 1/(2n - 1) )
= 0
Hence series is convergent
Learn More:
2. a) Test the convergence of the following series.1/6-1/11+1/16-4 ...
https://brainly.in/question/13373151
brainly.in/question/32296921
What is the greatest integer not exceeding the sum sigma (n=1 till ...
brainly.in/question/11760086