Which of the following sequences {an} converge and which diverge? Find the limit of each convergent Sequence? Sequence? an=2+(0.1)^n
Answers
(kqp-oscs-zgt) full explanation
Given : aₙ = 2 + (0.1)ⁿ
To Find : Sequence converges or diverges
Solution:
aₙ = 2 + (0.1)ⁿ
= 2 + (1/10)ⁿ
= (2 * 10ⁿ + 1 )/10ⁿ
aₙ₊₁ = (2 * 10ⁿ⁺¹ + 1 )/10ⁿ⁺¹
aₙ₊₁/aₙ = { (2 * 10ⁿ⁺¹ + 1 )/10ⁿ⁺¹}/{ (2 * 10ⁿ + 1 )/10ⁿ}
= (2 * 10ⁿ⁺¹ + 1 )/(10 (2 * 10ⁿ + 1 ))
= (2 * 10ⁿ⁺¹ + 1 ) / (2 * 10ⁿ⁺¹ + 10)
Denominator is greater than Numerator
Hence aₙ₊₁/aₙ < 1
Hence Series is convergent
Lim n→ ∞ aₙ
Lim n→ ∞ 2 + (1/10)ⁿ
= 2 + 0
= 2
convergent Sequence
Limit is 2
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