Determine the radius of convergence for the given power series. $\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }$
A) $R =$ 4
B) $R =$ $\infty$
C) $R = \frac { 1 } { 4 }$
D) $R =$ 0

Question

Quizplus · Accepted Answer

The radius of convergence $R$ for the power series $\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }$ can be found using the ratio test. The ratio test gives us $\lim _{ k \to \infty } \left| \frac { a_{k+1} } { a_k } \right| = \lim _{ k \to \infty } \left| \frac { (k+1) x ^ { k+1 } } { 4 ^ { k+2 } } \cdot \frac { 4 ^ { k+1 } } { k x ^ { k } } \right| = \lim _{ k \to \infty } \left| \frac { (k+1) } { k } \cdot \frac { x } { 4 } \right| = \left| \frac { x } { 4 } \right|$. For the series to converge, this limit must be less than 1, which gives us $|x| < 4$, hence $R = 4$.

Determine the Radius of Convergence for the Given Power Series ∑k=0∞kxk4k+1\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }∑k=0∞​4k+1kxk​

Determine the Radius of Convergence for the Given Power Series $\sum _ { k = 0 } ^ { \infty } \frac { k x ^ { k } } { 4 ^ { k + 1 } }$