A Sequence So Fine

Calculus Level 4

Given that $f(x) = 3 \cdot 4^x \cdot 5^{-2x}$ , evaluate the expression $\displaystyle \sum_{a=1}^{\infty} \; f\left (a + \prod_{b=1}^{\infty} f(b) \right ) \; + \; \displaystyle \prod_{c=1}^{\infty} \; f\left (c + \sum_{d=1}^{\infty} f(d) \right ).$

\frac{4}{7}

\frac{7}{25}

\frac{4}{25}

\frac{4}{5}

1 solution

Guilherme Dela Corte
Apr 8, 2014

$f(n) = 3 \cdot \left (\frac{4}{25} \right )^n (\text{Geometric Progression})$

$\displaystyle \sum_{m=1}^{\infty} f(m) = \tfrac{f(1)}{1 - \tfrac{f(2)}{f(1)}} = \frac{4}{7}$

$c \in \mathbb{R}, \lim_{n \to \infty} f(n + c) = 0 \Rightarrow \displaystyle \prod_{n=1}^{\infty} f(n) = 0$

$\boxed{\text{expression} = \frac{4}{7}}$

You need to be careful with using the same variable in both the summation, and the product. The question as stated isn't correct mathematically.

Edit: The question has been updated since this comment was made.

Calvin Lin Staff - 7 years, 2 months ago

Just corrected the question. Thanks for telling me that, Calvin! :)

Guilherme Dela Corte - 7 years, 2 months ago

A Sequence So Fine

1 solution

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