First test applied. If the Limit = 0, no info

(continue testing).

If the Limit ≠ 0, the series diverges.

This includes Limit=∞, or Limit D.N.E.

p series looks like (1/n)^p If p > 1, series converges

if p <= 1, series diverges

∑ = a_n, find a b_n that you recognize as C or D

If a_n < b_n , and b_n converges, then a_n also converges

If a_n > b_n , and b_n diverges, then a_n also diverges

∑ = a_n find a b_n by removing terms of least magnitude

Run convergence tests on  b_n 

Take the Limit of a_n / b_n 

If the limit>0,  a_n  acts the same as b_n

Three things must be demonstrated first:

1) The series is positive

2) Continuous

3) always decreasing. (proved by comparing a_n <a_n-1, or first derivative<1)

a_n D or C if:   Lim  ∫^t f(x) dx  C or D

        ^∞ 

The trick is to integrate the improper integral correctly.

LOOK FOR FACTORIALS!

 or ALTERNATING SERIES

Take the Limit as n→∞ of (a_n+1 / a_{n )}

• If L < 1  series converges

• If L > 1  or =∞  series diverges

• If L = 1 no info

LOOK POWERS OF n

 Add ⁿ√  (nth root)

Take the Limit as n→∞ of (ⁿ√a_{n )}

 • If L =0 no info 

• If L < 1 series converges

 • If L > 1 or =∞  series diverges

If you see an alternating series, try ratio test first

  a_n(-1)ⁿ

series converges if :

• If a_n+1 < a_n (Alway decreasing)

• If Limit as n→∞ of a_n=0  

Take the absolute value of a series:

∑ = |a_n | = a_n (usually alternating)

Test for C or D using appropriate test

if  |a_n | Converges, and   a_n  also Converges

the series is  ABSOLUTELY  Convergent  

if  |a_n | diverges, but  a_n  Converges

the series is CONDITIONALLY Convergent  

Recodgnize the power series: it has an X

ONE OF THE FOLLOWING IS TRUE:

• Power series converges only if x=0

• Series is absolutely convergent for every x 

• There exists an |r|>0 where the series is absolutely convergent. Divergent where x<-r or x>r.

Use the ratio test, should leave you with |x+c|<1

|x|<1 = r this is the radius of convergence. 

solve the inequality for x, these are the end points for the interval of convergence.

Test the endpoints for C or D using appropriate tests

Plug in x₁ & x₂ to the original series.

C: include the end point

D: exclude the end point 

Find the generall expression for the polynomial:

_n

Polynomial= ∑  f(c)  (x-c)ⁿ 

             ------

              (n)!

if the series {a_n} diverges, ( Limit of a_n DNE)

so does ∑a_n diverge, ( Limit ≠ 0 )

but

if the series {a_n} diverges, ( Limit of a_n exists)

MAYBE ∑a_n diverges ( Limit = 0 )