A Better Approximation at Lower Input Values

Question

I'm doing some research on an algorithm and to give an upper bound I wanted to simplify the term, call it $T$, below.

\begin{equation} T(i)=\sum_{k=1}^{i-1}\sec\Big(\big(\frac{3}{4}\big)^{k-1}\big(45^{\circ}\big)\Big), \textrm{ for } i\geq 2\end{equation}

I've been able to come up with the approximation $T \leq i$, and this approximation is good enough after the first few values of $i$, but the lower the value of $i$, the more accurate I need $T$ to be. Is there a more accurate approximation at lower values of $i$ for this?

Answer 1

Using that $\sec x<1+x^2/2$ you can get $$ T(i)<i-1+\frac{\pi^2}{126}\Bigl(9-16\Bigl(\frac{9}{16}\Bigr)^i\Bigr)=i-0.709042-1.25328\Bigl(\frac{9}{16}\Bigr)^i. $$ This can be improved using more terms in the Taylor series of $\sec x$ around $x=0$.