Find the sum of the series.
-$$x ^ { 5 } - x ^ { 7 } + x ^ { 8 } - x ^ { 4 } + \ldots$$
A) $\frac { 5 x } { 1 - x }$
B) $\frac { x ^ { 5 } } { 1 - x }$
C) $\frac { x ^ { 5 } } { 1 + x }$
D) $\frac { x ^ { 4 } } { 1 + x }$

Question

Quizplus · Accepted Answer

We can rewrite the given series as:
$$x^4-x^7+x^8-x^9+x^{12}-\ldots = x^4(1-x^3+x^4-x^5+x^8-\ldots)$$
Now, we recognize the expression within the parenthesis as an infinite geometric series with first term $1$ and common ratio $-x^3$. Thus, the sum of this series is:
$$\frac{1}{1+x^3}$$
Therefore, the sum of the original series is:
$$x^4 \cdot \frac{1}{1+x^3}= \boxed{\textbf{(C) } \frac{x^5}{1+x}}$$

Find the Sum of the Series x5−x7+x8−x4+…x ^ { 5 } - x ^ { 7 } + x ^ { 8 } - x ^ { 4 } + \ldotsx5−x7+x8−x4+…

Find the Sum of the Series $x ^ { 5 } - x ^ { 7 } + x ^ { 8 } - x ^ { 4 } + \ldots$