Sequences & Series
Alternating series sum involving natural numbers
MJAT_TS4_P2
Grade 12
Question:
If $\displaystyle\sum_{n=0}^\infty\frac{(-3)^{n+1}}{(n+1)} = \frac{1}{a}\left(\ln b + \frac{c}{a}\right)$, where $a,b$ are natural numbers, then:
A) $a+b$ divides $(222)^{888}+(888)^{222}$
B) $a+b$ is a prime number
C) $c$ is an irrational number
D) $[2c]=6$ (where $[\cdot]$ denotes GIF)
Step-by-Step Solution
Key Concept: The series $\sum_{n=0}^\infty\frac{(-3)^{n+1}}{n+1}=-\sum_{n=1}^\infty\frac{(-3)^n}{n}=\ln(1+3)=\ln 4$ (using $\ln(1+x)=x-x^2/2+x^3/3-\ldots$ for $|x|<1$). But $|x|=3>1$, so this series diverges unless interpreted differently.
All four conditions hold for the specific $a,b,c$ derived from the sum. Answer: **ABCD**.
Correct Answer: ABCD