Question

Consider the power series [infinity] 7n x n! n=1 Find the radius of convergence R. If it is infinite, type "infinity" Answer: R= What is the interval of convergence? Answer (in interval notation): → I- n or "inf".

Answer 1

To determine the radius of convergence, we can use the ratio test for power series. Let's apply the ratio test to the given power series:

```
lim(n→∞) |(7(n+1) x (n+1)!) / (7n x n!)|
= lim(n→∞) |7(n+1) x (n+1)! / (7n x n!)|
= lim(n→∞) |(7n + 7) x (n+1)! / (7n x n!)|
= lim(n→∞) |(n + 1) / n|
= 1
```

Since the limit of the absolute value of the ratio is equal to 1, the radius of convergence is infinite (R = ∞).

For the interval of convergence, since the radius of convergence is infinite, the series converges for all real numbers. Therefore, the interval of convergence is (-∞, +∞), which can be represented as "(-inf, inf)" in interval notation.