As the question says,
$$\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $$
where a is a constant, $a>0$.
2026-03-31 18:57:47.1774983467
Finding $\lim_{n\to \infty} \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} $
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Hint. We assume $\ln a \neq0$. One may use the Riemann sum result, as $n\to \infty$, giving $$ \frac{1}{n} \sum_{k=1}^n \frac{a^{1+\frac{k}{n}}}{a^{1+\frac{k}{n}}+1} \to\int_0^1\frac{a^{1+x}}{a^{1+x}+1}dx=\frac1{\ln a}\int_0^1\frac{(a^{1+x}+1)'}{a^{1+x}+1}dx. $$ Can you take it from here?