Sequences & Series
Sequence and Series
Allen Star Batch
Grade 11
Question:
Let $f(x) = \frac{1}{x+1} + \frac{2x}{(x+1)(x+2)} + \frac{3x^2}{(x+1)(x+2)(x+3)} + \cdots + \frac{nx^{n-1}}{(x+1)(x+2)(x+3)\cdots(x+n)}$, then:
$f(x) = \frac{x}{1+x} + \frac{x^n}{(x+1)(x+2)\cdots(x+n)}$
$1 - \frac{x^n}{(x+1)(x+2)\cdots(x+n)}$
$f'(x) = \left[-\frac{n}{(x+1)(x+2)\cdots(x+n)}\right]\left(\sum_{r=1}^{n} \frac{x}{x+r}\right)$
$f'(x) = \left[-\frac{x^{n-1}}{(x+1)(x+2)\cdots(x+n)}\right]\left(\sum_{r=1}^{n} \frac{x}{x+r}\right)$
Step-by-Step Solution
Key Concept: Use telescoping partial fractions and logarithmic differentiation to relate the derivative to a product formula.
Given $f(x) = \sum_{r=1}^{n} \frac{rx^{r-1}}{(x+1)(x+2)\cdots(x+r)}$, use partial fraction decomposition to express each term. Telescope the series by writing $\frac{x^{r-1}}{(x+1)(x+2)\cdots(x+r-1)} - \frac{x^r}{(x+1)(x+2)\cdots(x+r)}$. From $1-f(x) = \prod_{r=1}^{n} \frac{x}{r+x}$, derive $f'(x) = -\frac{x^{n-1}}{(x+1)(x+2)\cdots(x+n)} \sum_{r=1}^{n} \frac{r}{r+x}$.
Correct Answer: 2,4