Moderate -0.3 This question tests standard geometric series convergence conditions (|r| < 1) and sum to infinity formula. Part (a) requires identifying r = -2x and solving |-2x| < 1, while part (b) applies S = a/(1-r) = 1/(1-(-2x)) = 8. Both parts are routine applications of well-known formulas with straightforward algebraic manipulation, making this slightly easier than average.
15. An infinite geometric series has first four terms \(1 - 2 x + 4 x ^ { 2 } - 8 x ^ { 3 } + \cdots\). The series is convergent.
a. Find the set of possible values of \(x\) for which the series converges.
Given that \(\sum _ { r = 1 } ^ { \infty } ( - 2 x ) ^ { r - 1 } = 8\),
b. calculate the value of \(x\).
Find set of possible values of \(x\) for convergence of \(1 - 2x + 4x^2 - 8x^3 + \ldots\)
(2)
M1 Understands that for series to be convergent \(
Part (b):
Answer
Marks
Guidance
Calculate value of \(x\) given \(\sum_{r=1}^{\infty} (-2x)^{r-1} = 8\)
(3)
M1 Understands to use sum to infinity formula, e.g. \(\frac{1}{1+2x} = 8\); M1 Attempts to solve for \(x\), e.g. \(\frac{1}{8} = 1 + 2x \Rightarrow 2x = -\frac{7}{8} \Rightarrow x = \ldots\); A1 \(x = -\frac{7}{16}\)
**Part (a):**
Find set of possible values of $x$ for convergence of $1 - 2x + 4x^2 - 8x^3 + \ldots$ | (2) | M1 Understands that for series to be convergent $|r| < 1$ or states $|-2x| < 1$; A1 Correctly concludes that $|x| < \frac{1}{2}$. Accept $-\frac{1}{2} < x < \frac{1}{2}$
**Part (b):**
Calculate value of $x$ given $\sum_{r=1}^{\infty} (-2x)^{r-1} = 8$ | (3) | M1 Understands to use sum to infinity formula, e.g. $\frac{1}{1+2x} = 8$; M1 Attempts to solve for $x$, e.g. $\frac{1}{8} = 1 + 2x \Rightarrow 2x = -\frac{7}{8} \Rightarrow x = \ldots$; A1 $x = -\frac{7}{16}$
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15. An infinite geometric series has first four terms $1 - 2 x + 4 x ^ { 2 } - 8 x ^ { 3 } + \cdots$. The series is convergent.\\
a. Find the set of possible values of $x$ for which the series converges.
Given that $\sum _ { r = 1 } ^ { \infty } ( - 2 x ) ^ { r - 1 } = 8$,\\
b. calculate the value of $x$.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q15 [5]}}