PRESENTATION BY KARAN PANCHAL

SUBJECT : CALCULAS

TOPIC NAME : INTRODUCTION OF SEQUENCE

WHAT U A SEQUENCE

• A sequence is a list of numbers

in a given order.• Each a is a term of the sequence.• Example of a sequence:• 2,4,6,8,10,12,…,2n,…• n is called the index of an

1 2 3, , , , ,na a a a

• In the previous example, a general term an of index n in the sequence is described by the formula

an= 2n.• We denote the sequence in the previous

example by {an} = {2, 4,6,8,…}• In a sequence the order is important:• 2,4,6,8,… and …,8,6,4,2 are not the same

• Example 6: Applying theorem 3 to show that the sequence {21/n} converges to 0.• Taking an= 1/n, limn∞ an= 0 ≡ L• Define f(x)=2x. Note that f(x) is continuous on

x=L, and is defined for all x= an = 1/n• According to Theorem 3, • limn∞ f(an) = f(L)• LHS: limn∞ f(an) = limn∞ f(1/n) = limn∞ 21/n

• RHS = f(L) = 2L = 20 = 1• Equating LHS = RHS, we have limn∞ 21/n = 1• the sequence {21/n} converges to 1

• Example 7: Applying l’Hopital rule• Show that • Solution: The function is defined for x

≥ 1 and agrees with the sequence {an= lnn /n} for n ≥ 1.

• Applying l’Hopital rule on f(x):

• By virtue of Theorem 4,

lnlim 0n

nn

ln( ) xf xx

ln 1/ 1lim lim lim 01x x x

x xx x

lnlim 0 lim 0nx n

x ax

• Example 12 Nondecreasing sequence• (a) 1,2,3,4,…,n,…• (b) ½, 2/3, ¾, 4/5 , …,n/(n+1),… (nondecreasing because an+1-an ≥ 0)• (c) {3} = {3,3,3,…}

• Two kinds of nondecreasing sequences: bounded and non-bounded.

• Example 13 Applying the definition for boundedness• (a) 1,2,3,…,n,…has no upper bound• (b) ½, 2/3, ¾, 4/5 , …,n/(n+1),…is bounded

from above by M = 1.• Since no number less than 1 is an upper bound

for the sequence, so 1 is the least upper bound.