calc c - series and sequences

Harmonic series

1/n diverges

Geometric series

ar^n a series where all sequencial terms yeild a common ratio. Sn = a(1-r^n)/(1-r) so converges to a/(1-r) when -1<r<1

subtraction trick

write the nth partial sum generally (Sn = a1 + a2 + a3 + … + an) and then create another identical term but multiply both sides by some value and subtract. this will usually leave the first and end terms as a simple answer to nth partial sum

sequence of terms

a list of all the terms that will be added together in the series.

sequence of partial sums

a list of all the partial sums up to some n value (S5 is the sum of all the first 5 term where n=5). the limit of this sequence can tell us if the series converges or diverges

sequence convergence rules

1/n^p converges when p>0, r^n = {0 (-1<r<1), 1 (r=1), diverges otherwise}

sequence vs function rule

for a sequence that follows a continuous curve described by f(x) if the limit of f(x) = L then the limit of the sequence is L