Infinite Sequences and Series (BC Only)

Changing the exponent on n from 1 to any number greater than or equal to one

Changing the exponent on n from 1 to exactly 2

Keeping the exponent on n as is without change

Changing the exponent on n from 1 to any number less than 1

$t_\infty$ is a convergent series.

It cannot be determined whether $t_\infty$ is convergent or divergent.

$t_\infty$ is a divergent series.

$t_\infty$ is neither convergent nor divergent

Ratio Test

Integral Test

Root Test

P-Series Test

The terms of the series become zero as the number of terms increases.

The sum of its terms increases without bound.

The sum of its terms approaches a finite value.

Each term in the series is less than the one before it.

Replacing factorial with $(x-a)$ raised to a higher power maintaining differentiability of f.

Increasing n by one in all terms while keeping f differentiable at x = a.

Making f non-differentiable at every point near x = a.

Decreasing x by half towards approaching limit near center 'a'.

$p = 1$

$p \leq 0$

$p > 1$

$0 < p \leq 1$

It converges because $-1 < r < 1$

Not enough information is given to determine convergence or divergence without knowing the first term's value

It diverges because $r > -\frac{1}{2}$ only ensures divergence for positive ratios

It diverges because $r < -1$

Ratio Test

Root Test

Integral Test

p-Series Test

$\lim_{x \to \infty} \frac{f^{(a)}(xx)}{f^{(a)} + i(xx)n!}$

$\lim_{x \to \infty} \frac{|f^{(a)}(x)|}{|f^{(a)}(x+1)|}$

$\lim_{x \to \infty} \frac{f(f^{(a+n)}(x))}{f(n!(x))}$

$\lim_{x \to \infty} \frac{f^{(a)}(x)}{f(f^{(a)}(x))}$

The geometric series diverges due to oscillation between terms when raised to higher powers of n.

The geometric series neither converges nor diverges but approaches an asymptote as n increases indefinitely.

The geometric series converges because the absolute value of r is less than one.

Convergence cannot be determined without knowing the first term of the geometric series.