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

Question

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

Quizplus · Accepted Answer

The given series can be split into two geometric series: $1 + r^2 + r^4 + \ldots$ and $3r + 3r^3 + 3r^5 + \ldots$. Both are geometric series with common ratios of $r^2$ and $3r$, respectively. For a geometric series to converge, the absolute value of the common ratio must be less than 1. Thus, we need $|r^2| < 1$ and $|3r| < 1$. The first condition simplifies to $|r| < 1$, which also ensures the second condition is met, since if $|r| < 1$, then $|3r| < 3$, which is less than 1. Therefore, the series converges if $|r| < 1$.

Solve the Problem 1+3r+r2+3r3+r4+3r5+r6+…1 + 3 r + r ^ { 2 } + 3 r ^ { 3 } + r ^ { 4 } + 3 r ^ { 5 } + r ^ { 6 } + \ldots1+3r+r2+3r3+r4+3r5+r6+…

Solve the Problem $1 + 3 r + r ^ { 2 } + 3 r ^ { 3 } + r ^ { 4 } + 3 r ^ { 5 } + r ^ { 6 } + \ldots$