Calc Study Cards

Geometric

• -1 < r < 1 → Converges

• r < -1 or r > 1 → Diverges

• a value = whatever is under the sigma(eg. “n= 0” or “n=1”)

• An alternator in front of the series flips the sign of the every term in the series

Nth Term Test

• Only tells Divergence

• If the limit of the series:

  • = 0 → diverges

  • ≠ 0 → inconclusive

Integral Test

• Function has to be:

  • positive

  • decreasing

  • continuous

• If the the integral converges then the series converges and vise versa

• If the ending value:

  • any number (including zero) → converges

  • infinity or negative infinity → diverges

P - series

• If p > 1 → converges

• If negative infinity < p =< 1 → diverges

Harmonic Series

• Always diverges

Limit Comparison Test

• If the compared function diverges, original → diverges

• If the compared function converges, original → converges

• If the compared:

  • Converges, compared >= original

  • Diverges, original >=compared

Alternating Series Test

• only tells convergence

• the next term of the alternating series and the original series converge if:

  • limit as n→infinity of the series = 0

  • the next term =< previous (You can also plug in values to check if this is true)

Ratio Test

• If limit as n → infinity of |next/previous| < 1 → converges

• If limits as n→ infinity of |next/previous| > 1 → diverges

• If limits as n→ infinity of |next/previous| = 1 → inconclusive