Calctest3

lim n→ ∞ (ln(n)/n)

lim n→ ∞ (ln(n)/n)

=0 (for ln)

lim n→∞ n^(1/n)

=1

lim n→∞ x^(1/n)

=1 for x>0

lim n→∞ x^n

=0 for |x|<1

lim n→ ∞ (1+x/n)^n

=e^x

lim n→∞ (x^n)/(x!)

=0

a harmonic series looks like

∞

∑ 1/n

n=1

a harmonic series always

diverges

a geometric series looks like

∞

∑ r^n

n=0

a geometric series converges to

1/(1-r)

Geometric series converge when

|r|<1

Geometric series diverge when

|r|≥1

A p-series looks like

∞

∑ 1/(n^p)

n=1

p-series converge when

p >1

p-series diverge when

p<1

ratio test

p= lim n→∞ |(aₙ₊₁)/(aₙ)|

ratio test when p<1

series converges absolutely

ratio test when p>1

series diverges

ratio test when p=1

inconclusive

root test

p= lim n→∞ |aₙ|^(1/n)

root test when p<1

series converges absolutely

root test when p>1

series diverges

root test when p=1

inconclusive