a) You need to use the ratio test to evaluate the radius of the convergence of the given sequence, hence, you need to evaluate the limit `lim_(n->oo) |(a_(n+1))/(a_n)|` , where `a_n = (x^n*(n+1))/(3^n)` such that:
`lim_(n->oo) |((n+2)x^(n+1))/(3^(n+1))*(3^n/((n+1)x^n))|`
Reducing duplicate terms yields:
`lim_(n->oo) |((n+2)x)/3)*(1/(n+1))|`
You need to factor out `x/3,` such that:
`|x|*lim_(n->oo) ((n+2))/3)*(1/(n+1)) = |x/3|*lim_(n->oo) (n(1 + 2/n))/(n(1 + 1/n))`
Reducing duplicate factors yields:
`|x/3|*lim_(n->oo) (1 + 2/n)/(1 + 1/n) = |x/3|*1 = |x/3|`
By the ratio test yields:
- if `|x/3| < 1 => |x| < 3` , the series converges
- if `|x| > 3` , the series diverges
Hence the radius of convergence of series is `R = 3.`
b) You need to evaluate the interval of convergence, hence, you need to solve the absolute value inequality `|x| < 3` , such that:
`|x| < 3 => -3 < x < 3`
Hence, evaluating the interval of convergence yields `(-3,3).`