Find the Radius of Convergence and the Interval of Convergence $$\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$$

question 14

Multiple Choice

Find the radius of convergence and the interval of convergence of the power series.
$$\sum _ { n = 0 } ^ { \infty } \left( \frac { n x } { 6 } \right) ^ { n }$$

• A) \(R = 0 , I = \{ 0 \}\)
• B) \(R = \infty , I = ( - \infty , \infty )\)
• C) \(R = 6 , I = [ - 6,6 ]\)
• D) \(R = 6 , I = ( - 6,6 )\)

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Definitions:

Diarthrosis Joint

A type of synovial joint that permits a variety of movements, such as bending, extending, and rotation, exemplified by joints like the knee, elbow, and shoulder.

Hyperextension

An extension of a limb or part beyond its normal limit, which can potentially cause injury.

Popliteal Ligament

A strong, fibrous band of tissue connecting the femur to the tibia at the knee, crucial for knee joint stability.

Posterior Knee

The backside of the knee joint, an area that includes tendons, ligaments, and bursae important for knee movement and stability.