I have a sum given by,
$$(1) \quad | S_w=\sum_{n=1}^{w} 3^{n/2} \cdot \sin(3^n \cdot t) |$$
How do I find the value of $(1)$ asymptotically?
I can guess, using knowledge about the fractal dimension of weierstrass functions to get,
$$(2) \quad S_w \sim w^{1/2}$$
How do I do I find the sum without resorting to knowledge about the fractal dimension of tangentially related objects?
Also, I wish to have information about how to find asymptotic forms for other highly oscillatory series. Any references or methods would help.