a)∑an is convergentb)is convergentc)∑an is divergent as does not converges to 0d)∑an is divergentCorrect answer is option 'B'. Can you explain this answer?

Question

a)&sum;an is convergentb)is convergentc)&sum;an is divergent as  does not converges to 0d)&sum;an is divergentCorrect answer is option 'B'. Can you explain this answer?

Answer

Apply Cauchy condensation test.

Answer

B

Answer

2ⁿ

If ai > 0 for all i, then (1 + a1)(1 + a2)... (1 + an) > 1.

Since a1, a2, a3, ...an = 1, then (1 + a1)(1 + a2)... (1 + an) = (2)(2)...(2) = 2ⁿ.

Therefore, the greatest value of (1 + a1)(1 + a2)... (1 + an) is 2ⁿ.

Answer

Option d is correct
As there is direct result for the convergence an= 1/n(logn) ^p converges if p>1 and diverges if p<= 1=''>

Answer

Let the given expansion be f(n) f(n) = (1 + a1)(1 + a2)... (1 + an) Also given, a1 = a2 = a3 = ... = an = 1

Consider for n = 2 f(2) = (1 + a1)(1 + a2) f(2) = (1 + 1)(1 + 1) = 22

Consider for n = 5 f(5) = (1 + a1)(1 + a2)(1 + a3)(1 + a4)(1 + a5) f(5) = (1 + 1)(1 + 1)(1 + 1)(1 + 1)(1 + 1) = 25

Similary for n times, it is given as f(n) = (1 + 1)(1 + 1) ...... (1 + 1) = 2n (1 + a1)(1 + a2)... (1 + an) = 2n

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