Summary of Convergence Tests

• @@|r| < 1@@

• @@|r| = 0@@

• @@|r| = 1@@

• @@|r| \ge 1@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|>1@@ or @@\infty@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|=1@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|<1@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|=0@@

• @@\lim_{n\rightarrow\infty}a_n \ne 0@@

• @@\lim_{n\rightarrow\infty}a_n = 0@@

• @@a_n \text{ is a positive term}@@

• @@a_n \text{ is a decreasing sequence}@@

The series @@\sum_{n=1}^{\infty}a_n@@ will converge using the nth Term Test if ...

• The limit of the terms approaches a non-zero constant.

• The limit of the terms approaches zero, but the test is inconclusive.

• The series is a geometric series with a ratio less than one.

• Never! The nth Term Test cannot prove a series is convergent.

The series @@\sum_{n=1}^{\infty}a_n@@ will converge using the Integral Test if ...

• If @@a_n=f(n)@@ is positive, decreasing, and continuous, and @@\int_1^{\infty}f(x)dx@@ converges then, @@\sum_{n=1}^{\infty}a_n@@ converges

• If @@a_n=f(n)@@ is positive and increasing, and @@\int_1^{\infty}f(x)dx@@ diverges then, @@\sum_{n=1}^{\infty}a_n@@ converges

• If @@a_n=f(n)@@ is negative, decreasing, and continuous, and @@\int_1^{\infty}f(x)dx@@ converges then, @@\sum_{n=1}^{\infty}a_n@@ converges

• If @@a_n=f(n)@@ is positive, decreasing, and continuous, and @@\int_1^{\infty}f(x)dx@@ diverges then, @@\sum_{n=1}^{\infty}a_n@@ converges

The p-Series @@\sum_{n=1}^{\infty}\frac{1}{n^{\ p}}@@ will diverge if ...

• @@p > 1@@

• @@p = 0@@

• @@p < 1@@

• @@p = 2@@

The p-Series @@\sum_{n=1}^{\infty}\frac{1}{n^{\ p}}@@ will converge if ...

• p > 1

• p < 1

• p = 0

• p = 1

• If @@\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=L@@ and L is a finite number and @@L>0@@, then @@\sum_{n=1}^{\infty}a_n@@ and @@\sum_{n=1}^{\infty}b_n@@ both converge or both diverge.

• If @@\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=0@@, then @@\sum_{n=1}^{\infty}a_n@@ converges and @@\sum_{n=1}^{\infty}b_n@@ diverges.

• If @@\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=\infty@@, then @@\sum_{n=1}^{\infty}a_n@@ diverges and @@\sum_{n=1}^{\infty}b_n@@ converges.

• If @@\lim_{n\rightarrow\infty}\ \frac{a_n}{b_n}=1@@, then @@\sum_{n=1}^{\infty}a_n@@ and @@\sum_{n=1}^{\infty}b_n@@ both converge.

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|<1@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|=1@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|>1@@

• @@\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|\text{ does not exist}@@