The discussion revolves around evaluating the sum $\sum_{k=1}^{2013} f(k/2014)$, where the function $f(t)$ is defined as $f(t) := \frac{7^t}{7^t + \sqrt{7}}$. Participants are asked to perform this evaluation without computational tools.
There appears to be some agreement on the definition of the function and the task at hand, but the discussion includes corrections and acknowledgments of mistakes, indicating that the evaluation remains unresolved.
Participants have not provided a complete solution or detailed steps for the evaluation, and there may be assumptions or definitions that are not fully articulated.
magneto said:Define:
\[
f(t) := \frac{7^t}{7^t + \sqrt{7}}.
\]
Without the aid of a computer or calculator, evaluate:
\[
\sum_{k=1}^{2013} f \left( \frac{k}{2014} \right).
\]
(Please show the work.)
lfdahl said:$$\sum_{k=1}^{2013}f\left ( \frac{k}{2014} \right )= \sum_{k=1}^{1007}\left \{ f\left ( \frac{k}{2014} \right )+f\left ( \frac{2014-k}{2014} \right ) \right \}=^* \sum_{k=1}^{1007}1=1007$$
\[(*). \;\;\; f\left ( \frac{k}{2014} \right )+f\left ( \frac{2014-k}{2014} \right )=\frac{7^{\frac{k}{2014}}}{7^{\frac{k}{2014}}+7^{\frac{1}{2}}}+\frac{7^{\frac{2014-k}{2014}}}{7^{\frac{2014-k}{2014}}+7^{\frac{1}{2}}}\\\\ =\frac{7^{\frac{k}{2014}}\left ( 7^{\frac{2014-k}{2014}}+7^{\frac{1}{2} }\right )+7^{\frac{2014-k}{2014}}\left ( 7^{\frac{k}{2014}}+7^\frac{1}{2} \right )}{\left ( 7^{\frac{k}{2014}}+7^\frac{1}{2} \right )\left ( 7^{\frac{2014-k}{2014}}+7^\frac{1}{2} \right )} \\\\ =\frac{14 + 7^{\frac{1}{2}}\left ( 7^{\frac{2014-k}{2014}}+7^{\frac{k}{2014}}\right )}{14 + 7^{\frac{1}{2}}\left ( 7^{\frac{2014-k}{2014}}+7^{\frac{k}{2014}}\right )}=1\]