Determining Series Convergence and Divergence

These flashcards help review essential definitions and concepts related to determining series convergence and divergence based on various tests.

Divergence Test

If lim (n→∞) an does not exist (DNE) or is not 0, then Σ an is divergent.

p-Test

The series Σ a_n converges if p > 1 and diverges if p ≤ 1.

Integral Test

If f(x) is a continuous, positive, decreasing function, then Σ a_n converges if ∫ f(x) dx is convergent.

Comparison Test

If Σ an and Σ bn are series with positive terms, then if Σ an diverges, so does Σ bn; if Σ bn converges, then Σ an converges.

Limit Comparison Test

If lim (n→∞) an / bn = c, where c is a finite positive number, then both series Σ an and Σ bn converge or diverge together.

Alternating Series Test

If 0 ≤ a{n+1} ≤ an and lim (n→∞) an = 0, then the series Σ (-1)^n an converges.

Absolute Convergence Test

If Σ |an| converges, then Σ an converges absolutely.

Ratio Test

If lim |a{n+1}/an| = L, then the series converges if L < 1, diverges if L > 1, and is inconclusive if L = 1.

Conditional Convergence

A series is conditionally convergent if it converges but does not converge absolutely.

Divergence Conclusion

The divergence test cannot be used for series where the limit approaches zero; it is inconclusive.