Sequences & Series
Partial sums of geometric-like series with trig powers
MJAT_TS3_P1
Grade 12
Question:
For $x\in(0,\pi/4)$, define:
$$R_n=\sum_{r=1}^{2n}x^{4r-3},\quad S_n=\sum_{r=1}^{2n}x^{4r-2},\quad T_n=\sum_{r=1}^{2n}x^{4r-1},\quad U_n=\sum_{r=1}^{2n}x^{4r}$$
where $n\in\mathbb{N}$, $n\geq 4$. Then:
A) $R_n < S_n < T_n < U_n$
B) $R_n > S_n > T_n > U_n$
C) $\displaystyle\lim_{n\to\infty}(R_n+S_n+T_n+U_n)=\frac{x}{1-x}$
D) The value of $x$ for which $R_n+S_n=T_n+U_n$ is $2\sin\frac{\pi}{10}$
Step-by-Step Solution
Key Concept: Since $x\in(0,\pi/4)\subset(0,1)$: $x^1>x^2>x^3>x^4>\ldots$ So each term of $R_n$ dominates the corresponding term of $S_n$ etc. $R_n>S_n>T_n>U_n$ (B ✓, A ✗). As $n\to\infty$: all four sums combine to $\sum_{k=1}^\infty x^k=x/(1-x)$ (C ✓).
B ✓ (since $x<1$: $x^k$ decreasing). C ✓ ($\sum_{k=1}^\infty x^k=x/(1-x)$). D ✗. Answer: B, C.
Correct Answer: BC