Let . Which statement correctly describes the convergence of…

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let

T=\displaystyle\sum_{n=1}^{\infty}\frac{5^{n}}{6^{n}+n}

. Which statement correctly describes the convergence of

T

? (a)

T

diverges because the ratio test gives a limit of

\dfrac{6}{5}>1

. (b)

T

diverges because

\displaystyle\lim_{n\to\infty}\dfrac{5^{n}}{6^{n}+n}=\dfrac56\ne 0

. (c)

T

converges because, for sufficiently large

n

\dfrac{5^{n}}{6^{n}+n}<\left(\dfrac{5}{6}\right)^{n}

and

\displaystyle\sum\left(\dfrac{5}{6}\right)^{n}

converges. (d)

T

diverges because the

n^{\text{th}}

-root test gives

\sqrt[n]{\dfrac{5^{n}}{6^{n}+n}}=\dfrac56+1>1