Infinite Sequences and Series (BC Only)

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.

a_n = n * a_{n-1}

a_n = e^n

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

a_n = n!

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.

No convergence since each succeeding term increases due to logarithmic growth outpacing root extraction slowdown rate in $\ln(n)/\sqrt{n}$ sequence items' growth rates comparison effects.

Only convergence for sufficiently large n where natural log becomes insignificant against square root effectivity concerning diminishing trend within series progression seen here typically when comparing such functions' growth decline rates respectively.

Convergence occurs for all values of n starting from one since each succeeding term's magnitude decreases while approaching zero.

Convergence only occurs for even values of n as odd values result in increasing magnitudes of subsequent terms.

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.

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

1 + 2 + 3 + 4 + ...

1 - 2 + 3 - 4 + ...

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

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

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

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

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

The series must have a finite number of terms.

If the series tests convergent, then it must always satisfy $\theta_n > 0$.

Each term in the series needs to be less than the previous one by at least half.

The total sum of the series should not change sign with each new term added.

Predictive measures taken via test application secure divergence proof status however overlooking critical component relationship yielding faulty conclusion aftermath.

There found inconsistency suggesting divergence amidst test application due to confusing overlap regarding factorial exponential interplay intricacies involved herewith per case basis treatment necessity.

Resultant implicates given series converging absolutely because ratio between $b_k$s & corresponding $p_k$s diminishes down close-to-zero final point.

Despite test indications favorably pointing toward potential singular directional outcome upon deeper investigation reveals contradictory evidence advocating opposite assertion thereby nullifying initial presumption altogether.