Infinite Sequences and Series (BC Only)

$p > 1$

$p < 1$

$p = 1$

$0 < p < 1$

An integral that is difficult to solve analytically.

An integral that involves irrational numbers.

An integral with infinitely many terms.

An integral with infinite limits of integration or an integrand that is unbounded.

Series automatically diverges due to infrequent negative terms.

The Integral Test still applies but requires more careful analysis before making any assertions about convergence.

Cannot apply Integral Test as series does not have the required positivity property all throughout the considered range.

Applying Integral Test would result in an indeterminate conclusion due to occasional negative values.

The series clearly converges as manifested by the fact that $f(k)=\sqrt{k+\frac{31}{(k+11)^k}}$ decreases monotonically for $k \geq 0$ while the subsequent integral from 0 to infinity is found to be convergent.

The sum diverges specifically because it can be related to a power series which typically show divergence unless direct constraints on said powers are asserted, ensuring effective control on term growth or decrease over extension ranges.

The series diverges since the function does not evenly decrease across all values of $k$, but some points could potentially see an increase invalidating the robustness required for applying the integral test with confidence.

The sum coincidentally converges though not because the related function continuously decreases, but due primarily to terms approaching zero with large enough values of $k$, implying innate series decay without needing integration analysis.

Sum the first ten terms of $f(n)$ and extrapolate to predict convergence.

Evaluate $\int_{2}^{\infty} \frac{1}{x(\ln{x})^2},dx$ to see if it converges or diverges.

Apply L'Hôpital's Rule to $\lim_{n \to \infty} f(n)$ to test for convergence.

Calculate the limit of $f(n)$ as $n$ approaches infinity and use this to conclude about convergence.

The terms of the series must be summed over an interval [k, infinity)

The terms of the series must be positive.

The terms of the series must be decreasing.

The series must be a power series.

Integrate $\left( \frac{-1}{k} \right) e^{-k^4t}$ over k from one to infinity as t goes towards infinity.

Compute partial sums up through k-terms and assess their stability as k increases indefinitely.

Graph several partial sums versus time t and observe behavior patterns as more terms are included.

Differentiate $\left( \frac{-1}{k} \right) e^{-k^4t}$ with respect to t and analyze limits as t approaches zero.

The Integral Test cannot be applied to this series.

Divergent

The Integral Test is inconclusive

Convergent

$\sum_{n=1}^{\infty} \frac{1}{n^2 + 1}$

$\sum_{n=1}^{\infty} \frac{1}{n^3}$

$\sum_{n=1}^{\infty} \frac{1}{n!}$

$\sum_{n=1}^{\infty} \frac{1}{e^n}$

Comparison Test

Ratio Test

P-Series Test

Integral Test