Sequences/Series

how to find if sequence conv. or div.?

-if limit = finite real number → conv.

-if limit = infinity or -infinity → div.

limit of sequence with (something)^n

-lny = lim as n→infinity Nln(something)

-lim as n→infinity ln(something) / (1/N) and use L’ Hospital

-solve for y to get limit

limit of sequence that oscillates in numerator/denominator (ex. (1+(-1)^N) / N or cosN/N)

-squeeze theorem

-take limit of function when N is even and odd or trig function = -1 and 1 (lim as n→infinity 0 and lim as n→infinity 2/N or lim as n→infinity -1/N and lim as n→infinity 1/N)

-if both limits approach same number, the limit of the given sequence is also that number

geometric series

ar^i

abs. value of r < 1 → conv. and sum = first term / (1-r)

abs. value of r > or = 1 → div.

to have limit = a/(1-r), N must start at 0 (ex. limit when starting at N = 0 is a/(1-r) - term at N=0)

harmonic series

1/N and diverges

p test

1/N^p

p < or = 1 → div.

p > 1 → conv.

telescoping series

(1/N - 1/(N+1)) = (1-1/2)+(1/2-1/3)+(1/3-1/4)+…=1

nth term / divergence test

-take limit of series

-limit = 0 → might conv.

-limit not = 0 → div.

-broad first check

integral test

-convert series to f(x)

-f(x) must be cont. and dec.

-integral from N to infinity of f(x)dx = finite number → conv.

-integral from N to infinity of f(x)dx = infinity or -infinty → div.

-The value of the improper integral in the Integral Test is not equal to the sum of the series because an integral calculates the smooth, continuous area under a curve, whereas a series calculates the discrete sum of the areas of rectangular blocks

-good for stuff that can be easily integrated

-not good for exponentials or factorials

comparison test

-compare the base series and a sub-series (ex. 1/(N²+5) and 1/N²)

-find if sub-series conv. or div.

-if sub-series is bigger and conv. → conv.

-if sub-series is smaller and div. → div.

-good for poly/poly and trig

-bad for natural log

limit comparison test (LCT)

-do this if comparison test fails

-compare the base series and a sub-series (ex. 1/(N²+5) and 1/N²)

-find if sub-series conv. or div.

-find limit of series / sub-series

-if 0 < limit < infinity, series conv. if sub-series conv. and div. if sub-series div.

-good for poly/poly

-bad for trig

ratio test

-if L < 1 → conv.

-if L > 1 → div.

-if L = 1 → can’t tell

-good for exponentials and factorials

-bad for poly/poly

root test

-if L < 1 → conv.

-if L > 1 → div.

-if L = 1 → can’t tell

-good for exponentials

-bad for everything else