To find: The values of r for the convergent sequence.

The sequence is convergent when |r|<1 and divergent when |r|≥1 .

Given:

The sequence is an={nrn} . (1)

Result used:

The sequence {rn} converges when −1<r≤1 and diverges for other values of r>1 .

That is, limn→∞rn={0 if −1<r<11if r=1 .

Calculation:

The absolute value of an is

|an|=|nrn|=|n|⋅|rn|=n|rn|≥|rn|

Obtain the value of limn→∞an=limn→∞(nrn) .

limn→∞(nrn)=limn→∞(nr−n)

Apply L’Hospital’s rule and simplify the terms.

limn→∞(nrn)=limn→∞(ddn(n)ddn(r−n))=limn→∞(1(−lnr)r−n)=limn→∞(rn−lnr)

Notice that the sequence {nrn} is convergent to zero if |r|<1 and divergent if |r|≥1 .

Therefore, it can be concluded that the sequence is convergent when |r|<1 and divergent when |r|≥1 .