For what values of \(r\) is the sequence \(\left\{n r^{n}\right\}\) convergent?
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 .