Show the series 1. 1–1/2+1/2² – 1/2³ + .......................is convergence .

Show the series

1. 1–1/2+1/2² – 1/2³ + .......................is convergence .

Sol:

Given that:–

1–1/2 + 1/2² – 1/2³ +...................œ

From the Geometric series test,

Here, r = –1/3

                   r ≤ 1

Then the series is convergence by the Geometric series test.

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Note:

The Geometric series:–

1+r+r²+r³+ ..............∞ 

(i) Convergence if |r| < 1

(ii) divergence if r ≥ 1

(iii) oscillatory if r ≤ –1 

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2. Test the convergence

1+1/3+1/5+1/7+.............1/2n–1...........

Sol:

∑Un = ∑1/2n–1

∑VN = ∑1/n

Lim n→∞ Un/Vn   

= Lim n→∞ 1/2n–1 ÷ 1/n

= Lim n→∞ n/2n–1

= Lim n→∞ n/n(1 ÷ 2–1/n)

= 1/2 (finite and positive)

But ∑Vn = ∑ 1/n¹ , Here p = 1 

By p series test then ∑Vn is divergence

By comparison test Vn is divergence.

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