Series Tests

geometric series

a + a*r + a*r + a*r² + a*r³ ... converges to [a/1-r] convergent if r<1 divergent if r>=1 Strategy: write out the first n terms leaving exponents in place, find what r is, in other words find what is being multiplied by the previous term.

alternating series

b₁-b₂+b₃-b₄+b₅-b₆ where bn >= 0 ∑(-1)ⁿ bn .......if bn+1 <= bn (if decreasing) .......if lim {n→∞} bn = 0 then (-1)ⁿ bn converges

conditionally convergent

if a sum is convergent but is not absolutely convergent

integral test

if an = f(n) if ∫ {1,∞} f(x) converges then ∑an converges if ∫{1,∞} fx diverges then ∑an diverges

test for divergence

if lim {n→∞} an =/= 0 ......then ∑an diverges

limit comparison test

if lim {n→∞} an/bn = c , where 0<c<∞ and an,bn >= 0 for all n ......then ∑an converges if and only if ∑bn converges ......then ∑an diverges if and only if ∑bn diverges if a is sometimes positive and sometimes negative we can use ∑|an|

root test

if lim {n→∞} ⁿ√|an| if lim < 1, an conveges if lim >1 an diverges if lim = 1 you know nothing

comparison test

if ∑an and ∑bn are series with positive terms ........if ∑an <= ∑bn and ∑bn converges, ∑an converges ........if ∑an <= ∑bn and ∑an diverges, ∑bn diverges if the sum of a series is less than another series but the functions are about the same, then if the smaller one diverges the one above must also diverge, if the bigger one converges, then the smaller one must converge also ∑(n⁵+3n²+42)/(n⁶√(n)+1) ~ ∑n⁵/n⁶ ~ ∑1/n³/² which is a p-series

ratio test

lim |(an+1)/(an)| {n→∞} (for n+1, just subsitute in n+1 for every n) if L<1 absolutely convergent and convergent if L>1 divergent if L=1 you know nothing

absolutely convergent

the sum of the absolute value of an is finite

p-series

∑ {n=1,∞} 1/n^p convergent if p>1 divergent if p<=1