Infinite series, Divergence tests + formulas

when does an infinite series diverge

-when limit does not exist
-if limit is infinite

when do you use telescoping series

when consecutive terms cancel, usually requires partial fractions

first step for telescoping series

-partial fractions if necessary

-find the first couple terms and S_n

how to finish telescoping series

find S_n and compute the limit

general geometric series formula

when does geometric series converge

|r|<1

formula for sum of geometric series

a/1-r

what if for geometric series |r|≥1

series diverges

Divergence test

if Σa_n is convergent then lim a_n =0;

why is it called divergence test

because it tells you if limit an does NOT equal 0 then Σan is DIVERGENT

using divergence test, what is
lim a_n=0

Nothing, it is inconclusive

can divergence test prove convergence?

no, only divergence

conditions for integral test
let a_n=f(n)

f is
-positive
-decreasing
-continuous
for all x≥1

if this converges then Σa_n ___?

converges

if this diverges then Σa_n ___?

diverges

general p-series formula

if p>1 for p-series

it converges, otherwise it diverges

conv/div?
Σ1/n

diverge, because using the p-series, it must be p >1 and here p=1

conv/div
Σ1/√n

diverge, because using p-series it must be p>1 and here p=1/2

conv/div?
Σ1/n²

converge, because using p-series p>1 and here p=2

conditions for direct comparison test

Σa_n,Σb_n are positive series

Direct comparison test
if Σb_n converges then Σa_n

converges provided 0≤a_n≤b_n

Direct comparison test
if Σa_n diverges then Σb_n

diverges provided 0≤a_n≤b_n

what series are mostly used in comparison tests?

p-series and geometric

limit comparison test conditions

Σa_n,Σb_n are positive series let it be (photo)

case 1 for limit comparison

if 0<L<∞, then Σa_n,Σb_n both converge or both diverge

case 2 for limit comparison

if L=0 and Σb_n converges, then Σa_n also converges

case 3 for limit comparison

if L=∞ and Σb_n diverges, then Σa_n diverges

Theorem 1 (Limits of sequences

Theorem 2 (squeeze theorem for sequences)

theorem 3 (continuous functions)

Theorem 4 (Bounded Monotonic Sequences)