A language model architect is adjusting the dimensionality d of a model's embeddings, which affects the periods of the sine and cosine functions used for positional information. The period T_k for the k-th component is calculated using the formula: T_k = 2π * b^(2(k-1)/d), where b is a fixed base greater than 1. If the architect increases the dimensionality d while keeping the component index k (where k > 1) and base b constant, how will the period T_k be affected?

Calculating RoPE Component Period

Diagnosing RoPE Implementation Anomaly