Excel Auto-Fill a Series Without Mouse (Keyboard Only)
You're looking for "Fill-Down" or "Fill-Right".
Select the range by moving to the cell to start with... then Shift+↓ or Shift+→ to where you want to fill the contents to... and then press Ctrl+D (to fill-down) or Ctrl+R (to fill-right).
Unfortunately, there is no shortcut key for fill-left or fill-up.
You can use keyboard short cuts to get to anything that's on the Ribbon. Press Alt and then you can see the keystrokes next to each section. To bring up the Fill Series dialog box press Alt and then H F I S.
Use Shift + ↓ to highlight where you want the series to go - must include the filled in cell that you want to key off of.
Alt + H F I S
Shift + Tab F
Enter
(Inspired by the answers above and some trial and error)