lim

King Abdulaziz University

Department of Mathematics Math304 Sequence and Series

For the sequences there is only one test which is the limit.

Let {a

_n

}

^∞_n=1

be a sequence, then {a

_n

}

^∞_n=1

converges if

n→∞

lim a

_n

∈ R i.e. the limit exist and diverges if lim

n→∞

a

_n

does not exist.

Here are some of the famous limits and theorems:

1.

n→∞

lim sin n, lim

n→∞

cos n does not exist 2.

n→∞

lim tan

⁻¹

n = π 2 , lim

n→∞

ln n = ∞

3.

n→∞

lim r

ⁿ

=

 



0, if |r| < 1;

1, if r = 1;

DNE, if r = −1 or |r| > 1.

4.

n→∞

lim

³ 1 + a

n

´

_n

= e

^a

, lim

n→∞

√

n

n = lim

n→∞

n

¹ⁿ

= 1 5.

If lim

x→∞

f(x) = L, and f (n) = a

n

∀ n ∈ N, then lim

n→∞

a

n

= L 6.

If lim

n→∞

|a

_n

| = 0, then lim

n→∞

a

_n

= 0 7.

If b

n

≤ a

n

≤ c

n

, and lim

n→∞

b

n

= L = lim

n→∞

c

n

, then lim

n→∞

a

n

= L

8. Every bounded monotonic sequence is convergent.

King Abdulaziz University

Department of Mathematics Math304 Sequence and Series

Name of The Test Series Convergence or Divergence Comments

DIVERGENCE If lim_n→∞an= 0,

TEST P^∞

n=1an Diverges

if lim

n→∞

a

_n

6= 0.

then P^∞

n=1anmay

or may not be convergent.

Geometric (i)Converges if|r|<1

Series

P

∞ n=0

ar

ⁿ with P^∞

n=0arⁿ=_1−r^a . (ii) Diverges if|r| ≥1.

p-series (i) Converges ifp >1.

P∞ n=1

1 n^p

(ii) Diverges ifp≤1.

Comparison (i) Converges if P^∞

n=1

bn The comparison series

Test P^∞

n=1an converges and P^∞

n=1bnis 0< an≤bn for alln.

an>0 (ii) Diverges if P^∞

n=1

bn often geometric or

diverges and p−series.

0< bn≤an for alln.

Limit Comparison (i) Converges if P^∞

n=0bn The comparison series

Test P^∞

n=1

an converges and P^∞

n=1

bnis 0≤ lim

n→∞

a_n b_n <∞.

an>0 (ii) Diverges if P^∞

n=1bn often geometric or

diverges and p−series.

0<_n→∞lim ^a_bⁿ

n ≤ ∞.

Integral (i) Converges if fis positive, decreasing,

Test P^∞

n=1

an

∞R

1

f(x)dxconverges. continuous forx≥1, f(n) =an>0 (ii) Diverges if and the improper integral

∞R

1

f(x)dxdiverges. is easy to evaluate.

Ratio (i) Converges(absolutely) if Inconclusive if

Test P^∞

n=1an _n→∞lim

¯¯

¯^an+1_a

n

¯¯

¯<1 _n→∞lim

¯¯

¯^an+1_a

n

¯¯

¯= 1.

(ii) Diverges if Use this test ifaninvolves

n→∞lim

¯¯

¯^an+1_an

¯¯

¯>1 or(∞) factorials or a number to thenthpower.

Root (i) Converges(absolutely) if Inconclusive if

Test P^∞

n=1an _n→∞lim ⁿp

|an|<1 _n→∞lim ⁿp

|an|= 1.

(ii) Diverges if Use this test ifaninvolves

n→∞lim

np

|an|>1 or(∞) onlynthpower.

Alternating (i) Converges ifan≥an+1 Use this test for

Series P_∞

n=1(−1)ⁿan and lim

n→∞an= 0 alternating series Test an>0 (ii) Diverges if

n→∞lim an6= 0

Absolute (i) Converges(absolutely)if Use this test for

Convergence P^∞

n=1an

P∞

n=1|an|converges. series contains positive and

Test negative terms

(including alternating series)