Effectively extracting motions from video is a critical and long-standing problem for action recognition. This problem is very challenging because motions (i) do not have an explicit form, (ii) have various concepts such as displacement, velocity, and acceleration, and (iii) often contain noise caused by unstable pixels. Addressing these challenges, we propose the Taylor video, a new video format that highlights the dominant motions (e.g., a waving hand) in each of its frames named the Taylor frame. Taylor video is named after Taylor series, which approximates a function at a given point using important terms. In the scenario of videos, we define an implicit motion-extraction function which aims to extract motions from video temporal blocks. In these blocks, using the frames, the difference frames, and higher-order difference frames, we perform Taylor expansion to approximate this function at the starting frame. We show the summation of the higher-order terms in the Taylor series gives us dominant motion patterns, where static objects, small and unstable motions are removed. Experimentally, we show that Taylor videos are effective inputs to popular architectures including 2D CNNs, 3D CNNs, and transformers. When used individually, Taylor videos yield competitive action recognition accuracy compared to RGB videos and optical flow. When fused with RGB or optical flow videos, further accuracy improvement is achieved. Additionally, we apply Taylor video computation to human skeleton sequences, resulting in Taylor skeleton sequences that outperform the use of original skeletons for skeleton-based action recognition. Code is available at: https://github.com/LeiWangR/video-ar.