2. a) Test the convergence of the following series.
1/6-1/11+1/16-4/21+5/26-....
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)
Hence its an alternating series
(-1)ⁿ⁺¹aₙ
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
Lets check
Lim n→ ∞ aₙ
Lim n→ ∞ n/(5n+1)
Dividing numerator and denominator by n
Lim n→ ∞ 1/(5+1/n)
= 1/(5+0)
= 1/5
# 0
Hence series is divergent.
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