If the sum of the series [12-13+(1)/(2^2)-(1)/(23)+(1)/(3^2)+(1)/(2^3)-(1)/(2^23)+(1)/(23^…

Question

If the sum of the series
$$\frac12-\frac13+\frac{1}{2^2}-\frac{1}{2\cdot3}+\frac{1}{3^2}+\frac{1}{2^3}-\frac{1}{2^2\cdot3}+\frac{1}{2\cdot3^2}-\frac{1}{3^3}+\frac{1}{2^4}-\frac{1}{2^2\cdot3}+\frac{1}{2^2\cdot3^2}-\frac{1}{2\cdot3^2}+\frac{1}{3^4}+\cdots$$
is $\dfrac{\alpha}{\beta}$, where $\alpha,\beta$ are coprime, then $\alpha+3\beta$ is equal to \_\_\_\_\_\_\_\_.

Answer: 7

Accepted Answer

*(This series' exact grouping was heavily garbled in the source; following the source's own working:)* the sum evaluates to $P=\dfrac12$, giving $\alpha=1,\beta=2$.

$$\alpha+3\beta = 1+6 = 7.$$

Solutions · 1