Interval convergence ?

Question

how would i find the interval of convergence for the following problem

summation 1 to infinity [ n^2 (x-1)^n / 2^n ]

thank you

Answer 1

You would use the ratio test to find the interval. After simplifying, you would end up with

abs(x-1) / 2 * lim n-> oo ((n+1)/n)^2 < 1

The expression must be less than 1 for the sum to converge. You will find the limit becomes one. Solving for x, we find

-1 < x < 3

You will have to check the endpoints of the interval. This can be done by simply plugging in x = -1 and x = 3 into the summation. You will find that they both do not converge. Therefore, the interval is -1 < x < 3.

NOTES: abs(n) means to take the absolute value of the expression n.

lim n->oo f(n) means the limit of f(n) as n approaches infinity.