Introduction of sequence
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Transcript of Introduction of sequence
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.