Let:

$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$

$$I_n = \int_0^1 \mathrm{e}^{nf(x)}g(x)\mathrm{d}x$$

where $f(x)$ and $g(x)$ are smooth functions (assume they are differentiable up to any order required), $g(x) > 0$, and $n$ is a positive integer.

Note that the sum looks like a Riemann sum, but not quite, because there is an $n$ in the exponent.

Prove or disprove, that $R_n = |(S_n - I_n) / S_n| \rightarrow 0$ as $n \rightarrow \infty$. If true, characterize how fast $R_n$ decays as $n$ increases; is it exponentially fast?

Let:

$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$

$$I_n = \int_0^1 \mathrm{e}^{nf(x)}g(x)\mathrm{d}x$$

where $f(x)$ and $g(x)$ are smooth functions (assume they are differentiable up to any order required), $g(x) > 0$, and $n$ is a positive integer.

Note that the sum looks like a Riemann sum, but not quite, because there is an $n$ in the exponent.

Prove or disprove, that $R_n = |(S_n - c I_n) / S_n| \rightarrow 0$ as $n \rightarrow \infty$, for some constant $c$ that may depend on $f,g$ and/or its derivatives, but does not depend on $n$. If true, find the value of $c$ and characterize how fast $R_n$ decays as $n$ increases; is it exponentially fast?

Let:

$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$

$$I_n = \int_0^1 \mathrm{e}^{nf(x)}g(x)\mathrm{d}x$$

where $f(x)$ and $g(x)$ are smooth functions (assume they are differentiable up to any order required), $g(x) > 0$, and $n$ is a positive integer.

Note that the sum looks like a Riemann sum, but not quite, because there is an $n$ in the exponent.

Prove or disprove, that $R_n = |(S_n - I_n) / S_n| \rightarrow 0$ as $n \rightarrow \infty$. If true,characterize how $R_n$ decays as $n$ increases; is it exponentially fast?

Let:

$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$

$$I_n = \int_0^1 \mathrm{e}^{nf(x)}g(x)\mathrm{d}x$$

where $f(x)$ and $g(x)$ are smooth functions (assume they are differentiable up to any order required), $g(x) > 0$, and $n$ is a positive integer.

Note that the sum looks like a Riemann sum, but not quite, because there is an $n$ in the exponent.

Prove or disprove, that $R_n = |(S_n - I_n) / S_n| \rightarrow 0$ as $n \rightarrow \infty$. If true, characterize how fast $R_n$ decays as $n$ increases; is it exponentially fast?