I understand fully part a and b.
But for part c they just plug in 1 , why is that? Why didnt they plug in 2 or 3 because those are between -5 and 5?
Then how do they calculate the sum by plugging in 1?
Thanks
Do the arithmetic! You have already shown that the derivative of $\displaystyle \frac{1}{5- x}$ is $\displaystyle \frac{1}{(5- x)^2}$ and, that by differentiating the power series for $\displaystyle \frac{1}{5- x}$, $\displaystyle \frac{1}{(5- x)^2}= \frac{1}{5}\sum \frac{n}{5^{n+1}}x^{n-1}$.
Setting x= 1 in both of those you get $\displaystyle \frac{1}{4^2}= \frac{1}{5}\sum \frac{n}{5^{n+1}}$.
Multiply both sides of that equation by 5.