Question

Use the ratio test to determine whether the series is convergent or divergent. 2 4/2^2 8/3^2 16/4^2 ......

Answer 1

I assume the given series is

`\displaystyle 2 + \frac4{2^2} + \frac8{3^2} + (16)/(4^2) + \cdots = (2^1)/(1^2) + (2^2)/(2^2) + (2^3)/(3^2) + (2^4)/(4^2) + \cdots = \sum_(n=1)^\infty (2^n)/(n^2)`

By the ratio test, the series diverges, since

`\displaystyle \lim_(n\to\infty) \left|(2^(n+1))/((n+1)^2) \cdot (n^2)/(2^n)\right| = 2 \lim_(n\to\infty) (n^2)/((n+1)^2) = 2 > 1`