Examples

Radius of convergence

Find the radius of convergence of the series:

(a) (b)

Solution

(a) Ratio of terms:

Therefore and the series converges for .

---

(b) This power series skips the odd powers. Apply the ratio test to just the even powers:

Interval of convergence

Find the radius and interval of convergence of the following series:

(a) (b)

Solution

(a)

(1) Apply ratio test:

Therefore and thus:

Preliminary interval: .

---

(2) Check endpoints:

Check endpoint :

Check endpoint :

Final interval of convergence:

---

(b)

(1) Apply ratio test:

Therefore:

Preliminary interval:

---

(2) Check endpoints:

Check endpoint :

Check endpoint :

Final interval of convergence:

Radius and interval for a few series

Find the radius and interval of convergence of the following series:

(a) (b)

Interval of convergence - further examples

Find the interval of convergence of the following series.

(a) (b)

Solution

(a)

Ratio of terms:

Therefore and the preliminary interval is .

Check endpoints: diverges and also diverges.

Final interval is .

---

(b)

Ratio of terms:

Therefore:

Preliminary interval:

Check endpoints: converges but diverges.

Final interval of convergence: