The terms of a series are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning.
A) converges; Ratio Test
B) diverges; Ratio Test
C) diverges; Integral Test
D) converges; Alternating Series Test
E) converges; Root Test

Question

The terms of a series  are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning. 
A) converges; Ratio Test 
B) diverges; Ratio Test 
C) diverges; Integral Test 
D) converges; Alternating Series Test 
E) converges; Root Test

Quizplus · Accepted Answer

Since the terms are defined recursively, we can use the Ratio Test:
$$
\begin{aligned}
\lim_{n 	o \infty} \frac{a_{n+1}}{a_n} &= \lim_{n 	o \infty} \frac{11}{a_n} \
&= \lim_{n 	o \infty} \frac{11}{11e^{a_{n-1}}} \
&= \frac{1}{e^{a_{n-1}}}
\end{aligned}
$$
Since $a_1$ is a positive constant, we can see by induction that $a_n$ is positive for all $n$. Therefore, we can apply the Ratio Test:
$$
\begin{aligned}
\lim_{n 	o \infty} \frac{1}{e^{a_{n-1}}} &= 0 \quad 	ext{(since $e^{a_{n-1}}$ grows faster than any power of $n$)} \
\end{aligned}
$$
Since the limit is less than 1, the series diverges. Therefore, the answer is (B).

The Terms of a Series Are Defined Recursively