Infinite Sequences and Series (BC Only)

Check if the terms $b_n = \frac{n}{n^2+1}$ decrease monotonically for all $n$.

Determine whether $\lim_{n \to \infty} n(n^2+1) = 0$.

Calculate the sum of the first ten terms to estimate convergence.

Apply the Integral Test to see if it converges or diverges.

INCORRECT. Derive a test that verifies the conditionality based on partial sums of the series.

CORRECT. Show that $\lim_{n \to \infty} (-1)^n\frac{n}{3^2}$ equals zero and that each term $(-1)^n\frac{n}{3^2}$ is monotonically decreasing.

INCORRECT. Develop a new convergence criteria based on integration testing for alternating series.

INCORRECT. Show that $\sum |n^2/3^2|$ diverges.

Verify if ${(-1)}^n a_n$ is ultimately decreasing.

Verify if the terms are increasing without any bounds.

Verify if the terms are alternating between negative and positive values.

Verify if the sequence has a finite number of terms.

Calculate the ratio between successive terms and verify if it is less than one.

Compare the sum of the first five terms to see if they are less than previous ones.

Check whether each subsequent term decreases in absolute value and $\lim_{n \to \infty}$ approaches zero.

Estimate the total sum by adding several terms together.

a_n = n * a_{n-1}

a_n = e^n

a_n = (-1)^n * a_{n-1}

a_n = n!

1 - 1/2 + 1/4 - 1/8 + ...

1 + 2 + 3 + 4 + ...

1 - 2 + 3 - 4 + ...

1/2 + 1/4 + 1/8 + 1/16 + ...

Calculating the factorial of a number

Analyzing the limit of a geometric sequence

Determining the convergence or divergence of an alternating series

Finding the sum of an arithmetic sequence

The series converges by comparison to $e^{-5}$.

Since the ratio between successive terms increases to infinity, the series diverges.

The terms do not approach zero, so the series diverges.

Since the terms alternate and decrease, the series converges by the Alternating Series Test.

It always converges by default.

It might converge or diverge; other tests are needed.

It always diverges by default.

It is considered an oscillating series without further tests.

$\frac{A}{(x-3)} + \frac{B}{(x+5)}$

$(Ax+B)e^{(Cx+D)x}$

$(Ax+B)(Cx+D)$

$\frac{(Ax+B)}{(Cx+D)}$