Given the series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$ estimate the error in using the partial sum $s _ { 8 }$ by comparison with the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$.
A) $R _ { 8 } \leq 2.6130051$
B) $R _ { 8 } \geq 0.0000052$
C) $R _ { 8 } \leq 0.0000051$
D) $R _ { 8 } \geq 0.0000051$
E) $R _ { 8 } \leq 0.000005$

Question

Quizplus · Accepted Answer

The error $R_8$ is the remainder of the series after the 8th term, which can be estimated by the series $\sum _ { m = 9 } ^ { \infty } \frac { 1 } { 4 ^ { m } }$. This geometric series has a sum of $\frac{1/4^9}{1-1/4} = \frac{1}{3 \times 4^8}$, which is approximately 0.0000051.

Given the Series ∑m=1∞3m4m(3m+5)\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }∑m=1∞​4m(3m+5)3m​

Given the Series $\sum _ { m = 1 } ^ { \infty } \frac { 3 m } { 4 ^ { m } ( 3 m + 5 ) }$