Fractions are a pain

Calculus Level 5

Find the value of $1 - \frac{7}{12} + \frac{25}{288} - \frac{91}{5184} + \dots + (-1)^r \frac{3^r + 4^r}{r \times 12^r} + \dots$

(Note: When $r=0$ , $(-1)^r \frac{3^r + 4^r}{r \times12^r}$ would be but for the sake of the problem, it's defined to be $1$ here.)

The answer is 0.489.

1 solution

Josh Banister
Nov 15, 2015

$\begin{aligned} 1 + \sum_{r=1}^{\infty} (-1)^r \frac{3^r + 4^r}{r \times 12^r} &= 1 + \sum_{r=1}^{\infty} (-1)^r (\frac{1}{r \times 4^r} + \frac{1}{r \times 3^r}) \\ &= 1 - \sum_{r=1}^{\infty} -(-1)^r \frac{( \frac{1}{3})^r )}{r} - \sum_{r=1}^{\infty} -(-1)^r \frac{(\frac{1}{4})^r}{r} \\ &= 1 - \ln (1 + \frac{1}{3}) - \ln (1 + \frac{1}{4}) \\ &= 1 - \ln(\frac{4}{3}) - \ln(\frac{5}{4}) \\ &= 1 - \ln (\frac{5}{3}) \\ &\approx 0.489 \end{aligned}$

Fractions are a pain

The answer is 0.489.

1 solution

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