, 2020

Solution:
Let $S_{k}=\frac{1}{5^{k}}$. In order to get the coefficient of $a_{2}$ to be $-1$, we need to have $S_{1}-3 S_{3}$. This subtraction makes the coefficient of $a_{3}$ become $-6$. Therefore, we need to add $7 S_{3}$ to make the coefficient of $a_{4}$ equal to $1$. The coefficient of $a_{4}$ in $S_{1}-3 S_{3}+7 S_{5}$ is $14$, so we must subtract $15 S_{4}$. We can continue the pattern to get that we want to compute $S_{1}-3 S_{2}+7 S_{3}-15 S_{4}+31 S_{5}-\cdots$. To prove that this alternating sum equals $a_{1}-a_{2}+a_{3}-a_{4}+\cdots$, it suffices to show
$$\sum_{i=1}^{n}\left(-(-2)^{i}+(-1)^{i}\right)\binom{n}{i}=(-1)^{i+1}$$
To see this is true, note that the left hand side equals $-(1-2)^{i}+(1-1)^{i}=(-1)^{i+1}$ by binomial expansion. (We may rearrange the sums since the positivity of the $a_{i}$'s guarantee absolute convergence.) Now, all that is left to do is to compute
$$$$
$\sum_{i=1}^{\infty} \frac{\left(2^{i}-1\right)(-1)^{i-1}}{5^{i}}=\sum_{i=1}^{\infty} \frac{(-1)^{i}}{5^{i}}-\sum_{i=1}^{\infty} \frac{(-2)^{i}}{5^{i}}=\frac{\frac{-1}{5}}{1-\frac{-1}{5}}-\frac{\frac{-2}{5}}{1-\frac{-2}{5}}=\frac{5}{42}$