Defining Convergent and Divergent Infinite Series - AP Calculus BC

A test using integrals to determine series convergence. Relates series convergence to improper integrals.

A series $ \sum a_n $ is absolutely convergent if $ \sum |a_n| $ converges. Stronger condition than simple convergence.

A test comparing limits of two series to determine convergence. Compares behavior of two series at infinity.

The terms decrease in absolute value and approach zero. Both conditions must hold for convergence.

Convergent. It is a p-series with $p = 1.5 > 1$. P-series converges when exponent exceeds 1.

$1$. It is a geometric series with $a = \frac{1}{2}$, $r = \frac{1}{2}$. Using $S = \frac{a}{1-r} = \frac{1/2}{1-1/2} = 1$.

The series does not approach a finite limit. Partial sums fail to approach any finite value.

If $\text{∫} f(x)dx$ converges, then $\text{∑} a_n$ converges. Same convergence behavior as corresponding integral.

A test using $\frac{a_{n+1}}{a_n}$ to determine convergence. Uses consecutive term ratios to test convergence.

Convergent. It is an alternating series. Terms decrease and approach zero, satisfying AST.

$S_n = 1 + \frac{1}{2} + \frac{1}{3} + \text{...} + \frac{1}{n}$. Sum of first $n$ terms of the harmonic series.

Convergent. It is the exponential series. Factorial growth ensures rapid convergence.

If $\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1$. Limit less than 1 indicates series convergence.

The finite value the series approaches. This limit exists and is finite for convergent series.

The common ratio $|r| < 1$. This ensures each term becomes smaller than the previous.

Divergent. It is known as the harmonic series. Classic example of divergence despite decreasing terms.

Converges if $p > 1$. Critical threshold separating convergence from divergence.

Divergent. Proven fact about this important series.

Convergent. It is a geometric series with $r = \frac{1}{2}$. Since $|\frac{1}{2}| < 1$, the series converges.

Conditional convergence: series converges, but absolute value does not. Absolute convergence implies regular convergence.

A series where intermediate terms cancel out. Partial fractions create canceling patterns.

A sum of infinitely many terms, expressed as $a_1 + a_2 + a_3 + \text{...}$. The notation shows an endless sum of terms.

A test to determine convergence of series with alternating signs. Applies to series of form $\sum (-1)^n a_n$.

The series $1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...}$. Famous divergent series despite terms approaching zero.

Comparison Test or Limit Comparison Test. Best suited for rational function series.

Convergent. It satisfies the Alternating Series Test. Terms decrease in magnitude and approach zero.

The sum of the first $n$ terms of a series. Building block for determining series convergence.

A series that does not approach a finite limit. Partial sums either grow without bound or oscillate.

A series that approaches a finite limit as more terms are added. Partial sums approach a specific value as $n \to \infty$.