Series and Sequences Equations

nth term of arithmetic sequence

a_n=a₁+d(n-1)

recursive representation given a₁ and d (arithmetic)

a_n=a_n-1+d

Sum of Arithmetic Sequence

S_n=n/2 (a₁+a_n)

Arithmetic sequence convergent or divergent?

Divergent

nth term of a geometric sequence

a_n=a₁r^n-1

recursive representation given a₁ and r (geometric sequence)

a_n = ra_n-1

Sum of geometric sequence

S_n = a1[(1-rn)/(1-r)]

Geometric sequence convergent or divergent

If absolute value of r ≥ 1 than divergent, if absolute value of r < 1 than convergent

nth term of a harmonic sequence

a_n = 1/n

Sum of harmonic sequence

S_n = 1 + ½ + 1/3 + ¼ + … 1/n

harmonic sequence convergent or divergent

divergent

nth term of a quadratic sequence

y_n = an²+bn+c

quadratic sequences convergent or divergent

divergent

how to tell if a sequence is quadratic

if the 2nd difference is constant than the sequence is quadratic

nth term of a telescoping sequence

a_n = (1)/(n²+bn+c)

telescoping sequence convergent or divergent

convergent

sum of elements of a telescoping sequence

S_n= sigma(1/(n-a) - 1/(n-b)