Series Convergence Tests

P-series

Converges if p>1

Diverges if p<=1

Harmonic series

Converges to a/(1-r) if -1<r<1

Divergent for r<=-1 or r<=1

Integral test

If the function is continuous, positive, and decreasing:

The integral between the bounds and the sum have the same behavior

Comparison test

If An >Bn: if Bn diverges, An diverges

If An<Bn: if Bn converges, An converges

Limit comparison test

If lim (n→infinity) An/Bn >0

Sum of An and Sum of Bn have the same behavior

Ratio Test

If lim (n→infinity) |A(n+1)/An|

>1 diverges

=1 inconclusive

<1 converges

Alternating Series Test

If An is decreasing

And lim(n→infinity) An = 0

Series converges