Math, asked by anwesha2015banerjee, 3 months ago

convergence of series 1/6-2/11+3/16-4/21+...plz give the answer

Answers

Answered by amitnrw
3

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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