Question

Use the Ratio Test to determine if the series converges. [infinity]Σ 5k^2/4^kk=1Select the correct choice below and fill in the answer box to complete your choice. A. The series diverges because r=_______B. The series converges because r=_____. C. The Ratio Test is inconclusive because r=_______

Answer 1

Answer:

B.The series converges because r=1/4

Step-by-step explanation:

We are given that

$\sum_{k=1}^{\infty}\frac{5k^2}{4^k}$

We have to find the  correct option.

Ratio test :$\lim_{k\rightarrow\infty} \mid \frac{a_{k+1}}{a_k}\mid =r$

If r< 1 then series convergent

If r>1 then the series divergent

If r=1 , test fails

$r=lim_{k\rightarrow \infty}\mid\frac{\frac{5(k+1)^2}{4^{k+1}}}{\frac{5k^2}{4^k}}}\mid$

$r=\lim_{k\rightarrow \infty}\mid{\frac{5(k+1)^2}{4^k\cdot 4}\times\frac{4^k}{5k^2}}\mid$

$r=\lim_{k\rightarrow \infty}\mid{\frac{(k+1)^2}{4(k^2)}\mid$

$r=\lim_{k\rightarrow \infty}\mid\frac{k^2(1+\frac{1}{k})^2}{4k^2}\mid$

$r=\frac{1}{4}$

r<1

Therefore, the series converges .

B.The series converges because r=1/4