How do you test the series Sigma n^-n from n is [1,oo) for convergence?
1 Answer
Using the ratio test
Explanation:
The ratio test finds the ratio between terms
Hence, in order to test
k->oo abs(u_(k+1)/u_k)
= k->ooabs((k+1)^-(k+1)/k^-k)
=k->oo abs(k^k/(k+1)^(k+1))
=k->oo abs(k^k/(k+1)^k*1/(k+1))
=k->oo abs((k/(k+1))^k*1/(k+1))
=k->oo abs((1/(1+1/k))^k*1/(k+1))
because k->oo 1/k=0
therefore k->ooabs((1/(1+1/k))^k)=k->ooabs((1/(1+0))^k)=abs(1)=1
therefore k->ooabs((1/(1+1/k))^k*1/(k+1))=k->ooabs(1*1/(k+1))
=k->ooabs(1/(k+1))=0
i.e.