Find the radius of convergence, R, of the series.

Question

∑(from n=2 to ∞) ((x+7)^n)/(7^n ln (n))

Find the interval, I, of convergence of the series. (Enter your answer using interval notation.)

Answer 1

Let

an=(x+7)n7nln⁡(n).a_n = \frac{(x+7)^n}{7^n \ln(n)}.an​=7nln(n)(x+7)n​.

The ratio test involves finding the limit of the absolute value of the ratio of successive terms:

L=lim⁡n→∞∣an+1an∣.L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|.L=n→∞lim​​an​an+1​​​.

Let's compute this ratio:

an+1=(x+7)n+17n+1ln⁡(n+1).a_{n+1} = \frac{(x+7)^{n+1}}{7^{n+1} \ln(n+1)}.an+1​=7n+1ln(n+1)(x+7)n+1​.

So,

∣an+1an∣=∣(x+7)n+17n+1ln⁡(n+1)⋅7nln⁡(n)(x+7)n∣.\left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(x+7)^{n+1}}{7^{n+1} \ln(n+1)} \cdot \frac{7^n \ln(n)}{(x+7)^n} \right|.​an​an+1​​​=​7n+1ln(n+1)(x+7)n+1​⋅(x+7)n7nln(n)​​.

Next you need to find out where L <1