+148 A series is given by 1/3^1+2/3^2+3/3^3+4/3^4+5/3^5+....
Francine correctly applies the ratio test to determine whether the series converges or diverges.
Which statement reflects Francine's conclusion?
From the ratio test, r = 3. The series converges.
From the ratio test, r=1/3. The series diverges.
From the ratio test, r = 3. The series diverges.
From the ratio test, r=1/3. The series converges.
+23246 The nth term is n / 3n.
The (n + 1)th term is (n + 1) / 3n + 1.
The ratio test: lim | [ (n + 1) / 3n + 1 ] / [ n / 3n ] |
= lim | [ (n + 1) / n ] · [ 3n / 3n + 1 ] |
= lim | [ (n + 1) / n ] · [ 1/3 ] |
= (1/3) · lim | (n + 1) / n |
= (1/3) · lim | ( n/n ) + (1/n) |
= (1/3) · lim | 1 + (1/n) |
= (1/3) · [ lim | 1 | + lim | 1/n |
= (1/3) · [ 1 + 0 |
= (1/3) <--- converges
