Root and Ratio Test

I.  Quiz

II.  Homework

III.  The Ratio Test

Theorem:  The Ratio Test

Let sum a_n be a series then

1)  If lim |a_n+1/a_n| < 1 then the series converges absolutely

2)  If   |a_n+1/a_n| >1 then the series diverges

3)  |a_n+1/a_n| = 1 then try another test.

Proof:  Suppose that  lim |a_n+1/a_n| <= R < 1 then for the tail,

a_n+1 < Ra_n

a_n+2 < Ra_n+1 < R²a_n  ...

a_n+k < R^ka_n

So that

sum a_n  < sum aR^k

Which converges by the GST.

Example Determine the convergence or divergence of

sum 4ⁿ/n!

We use the Ratio Test:

lim 4ⁿ⁺¹/(n+1)!/4ⁿ/n! = lim [n!/(n+1)!][4ⁿ⁺¹/4ⁿ = lim 4/(n + 1) = 0

Hence the series converges by the Ratio Test

Exercises  Determine the convergence or divergence of

A)  sum n!/[n(3ⁿ)]

B)  sum 2ⁿ/3ⁿ⁺¹ 

IV.  The Root Test

Let sum a_n be a series with nonzero terms at the tail, then

1) sum a_n converges absolutely if lim n^th root(a_n) < 1

2)  sum a_n diverges if lim n^th root(a_n) > 1

3)  if  lim n^th root(a_n) = 1 then try another test.

Example Determine the convergence or divergence of

sum [3n/(n + 3)]ⁿ

We use the root test:   lim n^th root [3n/(n + 3)]ⁿ  = lim 3n/(n + 3) = lim 3/1 = 3

Hence the series diverges.

Exercises Determine the convergence or divergence of

A)  sum exp[-n³ - nⁿ²]

B)  sum (n/(n + 1))ⁿ