Solve the problem.
-For what value of $r$ does the infinite series $1 + 3 r + 2 r ^ { 2 } + 3 r ^ { 3 } + r ^ { 4 } + 3 r ^ { 5 } + 2 r ^ { 6 } + 3 r ^ { 7 } + r ^ { 8 } + \ldots$ converge?
A) $| r |

Question

Solve the problem.
-For what value of $r$ does the infinite series $1 + 3 r + 2 r ^ { 2 } + 3 r ^ { 3 } + r ^ { 4 } + 3 r ^ { 5 } + 2 r ^ { 6 } + 3 r ^ { 7 } + r ^ { 8 } + \ldots$ converge?
A) $| r | < \frac { 3 } { 2 }$
B) $| r | < \frac { 5 } { 2 }$
C) $| r | < \frac { 2 } { 3 }$
D) $| r | < 1$

Quizplus · Accepted Answer

The given series does not follow a simple pattern that fits the criteria for geometric or arithmetic series, but the convergence of any infinite series depends on the absolute value of the ratio between successive terms getting smaller, specifically it must be less than 1 for the series to converge. This is a fundamental requirement for the convergence of an infinite series, regardless of its specific pattern. Therefore, for the series to converge, the condition $|r| < 1$ must be met.

Solve the Problem rrr Does the Infinite Series 1+3r+2r2+3r3+r4+3r5+2r6+3r7+r8+…1 + 3 r + 2 r ^ { 2 } + 3 r ^ { 3 } + r ^ { 4 } + 3 r ^ { 5 } + 2 r ^ { 6 } + 3 r ^ { 7 } + r ^ { 8 } + \ldots1+3r+2r2+3r3+r4+3r5+2r6+3r7+r8+…

Solve the Problem $r$ Does the Infinite Series $1 + 3 r + 2 r ^ { 2 } + 3 r ^ { 3 } + r ^ { 4 } + 3 r ^ { 5 } + 2 r ^ { 6 } + 3 r ^ { 7 } + r ^ { 8 } + \ldots$