Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } e ^ { - 6 n } ?$
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C) $\lim _ { n \rightarrow \infty } e ^ { - 6 n } \neq 0$
D)It is convergent.
E)It is divergent.

Question

Which of the following is true for the infinite series $\sum _ { n = 1 } ^ { \infty } e ^ { - 6 n } ?$ 
A)Its sequence of partial sums is not bounded from above.
B)Its sequence of partial sums is decreasing.
C) $\lim _ { n \rightarrow \infty } e ^ { - 6 n } \neq 0$
D)It is convergent.
E)It is divergent.

Quizplus · Accepted Answer

The given series is a geometric series with the common ratio $r = e^{-6}$, which is between -1 and 1. Therefore, the series converges, and its sequence of partial sums is bounded and increasing, not decreasing. The limit of $e^{-6n}$ as $n$ approaches infinity is 0, which satisfies the necessary condition for convergence of an infinite series.

Which of the Following Is True for the Infinite Series ∑n=1∞e−6n?\sum _ { n = 1 } ^ { \infty } e ^ { - 6 n } ?∑n=1∞​e−6n?

Which of the Following Is True for the Infinite Series $\sum _ { n = 1 } ^ { \infty } e ^ { - 6 n } ?$