This is a question of an exam on Numerical Analysis I had:
Consider the floating point system of base $2$, maximum number of decimals $53$, maximum exponent $1025$ and minimum exponent $-1022$. That is, the numbers in this system are of the form $$0.d_1d_2\ldots d_{53}\cdot 2^{e},\ -1022\leq e\leq 1025.$$ Calculate the value of the sum $$\sum_{k=0}^{1021}\frac{1}{4^k}$$ in this number system if we round it up and if we truncate it.
My attempt:
We have that $(1/4)_{10}=0.01_2$, where the subscript means the base used. Therefore, the sum in this floating number becomes \begin{align*} 1_{10}+(1/4)_{10}+(1/4^2)_{10}+(1/4^3)_{10}+\cdots&=1_2+0.01_2+0.0001_2+0.000001_2+\cdots\\ &=1.01010101\ldots_2\\ &=0.1010101\ldots 010\color{red}{1}\cdot 2^1. \end{align*} I think $\color{red}{d_{53}}=\color{red}{1}$ because the $1$'s are in the odd digits (and $53$ is odd). Since the following $1/4^k$ that doesn't add to the sum has a $1$ in the $55$th decimal, it makes no difference if we round up or if we truncate the sum.
Is my approach correct? Thanks!