The Ratio Test involves a sequence ak such that ak 6 0 for a

The Ratio Test involves a sequence {ak} such that ak 6= 0 for all k and ` = lim k |ak+1| |ak| . Show that The Ratio Test fails when ` = 1.

Solution

Simply consider the sequence a[k]=1 for all k

the limit a[k+1]/a[k]=1 for all k.

So the ratio is 1 .

Clearly the series 1+1+.....diverges , as the sequence of partial sums S[n]=n is unbounded.

So the ratio test fails when f=1