convergence of series 1/6-2/11+3/16-4/21+...plz give the answer
Answers
Given : 1/6-2/11+3/16-4/21+...
To Find : Convergent or Not
Solution:
1/6 - 2/11 + 3/16 - 4/21 + ....
= (-1)¹⁺¹.1/(5*1 + 1) + (-1)²⁺¹.2/(5*2 + 1) + (-1)³⁺¹.3/(5*3 + 1) + (-1)⁴⁺¹.⁴/(5*⁴ + 1) + .....
aₙ = (-1)ⁿ⁺¹.n/(5n + 1)
Hence Series is
∑(-1)ⁿ⁺¹.n/(5n + 1)
aₙ = (-1)ⁿ⁺¹.n/(5n + 1)
aₙ₊₁= (-1)ⁿ⁺².(n+1)/(5n + 6)
Ratio Test
L = Lim n→ ∞ | aₙ₊₁ /aₙ|
= | - (n+1)(5n + 1) /n(5n + 6) |
= | -(5n² + 6n + 1) (5n² + 6n) |
Dividing numerator and denominator by n²
= | -(5 + 6/n + 1/n²)/( 5 + 6/n)
n → ∞ hence 1/n² and 1/n = 0
= | -(5 + 0 + 0 )/(5 + 0)|
= | -5/5|
= | -1 |
= 1
as L = 1 no conclusion can be made about convergence or divergence
if L < 1 then convergent if L > 1 then divergent
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