Final answer:
If sequences an and bn are convergent and the limit of bn is a nonzero value L, then lim (1/bₙ) = 1/L, asserting that the sequence 1/bₙ is also convergent to the reciprocal of L.
Step-by-step explanation:
If sequences aₙ and bₙ are convergent, to draw conclusions about lim (1/bₙ), we must first deduce the limit that bn converges to. Let's assume that lim bₙ = L, where L is a nonzero real number. Since bₙ converges to a nonzero limit, we can say that for n sufficiently large, bₙ is nonzero and the sequence 1/bₙ is well-defined.
Now, if L ≠ 0, by the properties of limits, we can assert that lim (1/bₙ) = 1/L. It is crucial that L ≠ 0 because if L were zero, the sequence 1/bₙ would not converge as it would involve division by zero as n approaches infinity, which is undefined.
Therefore, provided that the limit of bn is a nonzero value, we can deduce the limit of 1/bₙ is simply the reciprocal of the limit of bn.