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Limits of Sequences of Real Numbers�

Mr. V. V. Chandavale

Asst. Prof

Department of Mathematics

Raje Ramrao Mahavidyalaya, Jath

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Sequences of Numbers

Definition

Examples

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Index

FAQ

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Limits of Sequences

Definition

Examples

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If a sequence has a finite limit, then we say that the sequence is convergent or that it converges. Otherwise it diverges and is divergent.

0

Index

FAQ

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0

Index

FAQ

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Limits of Sequences

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Notation

The sequence (1,-2,3,-4,…) diverges.

Index

FAQ

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Computing Limits of Sequences (1)

Index

FAQ

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Computing Limits of Sequences (1)

Examples

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2

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n2

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Index

FAQ

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Computing Limits of Sequences

Examples continued

3

Index

FAQ

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Formal Definition of Limits of Sequences

Definition

Example

Index

FAQ

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Immediate consequence of the formal definition of a sequence

  • Every convergent sequence is bounded.

Theorem

Proof

Suppose that lim xn=L . Take ϵ = 1 (any number works). Find N 1 so that whenever �n > N1 we have xn within 1 of L. Then apart from the finite set { a1, a2, ... , aN} all the terms of the sequence are bounded by L+ 1 and L - 1. �So an upper bound for the sequence is max {x1 , x2 , ... , xN , L+ 1 }. Similarly one can find a lower bound.���

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Index

FAQ

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The Limit of a Sequence is UNIQUE

  • The limit of a sequence is UNIQUE

Theorem

Proof

  • Indirectly, suppose, that a sequence would have 2 limits, L1 and L2. Than for a given ε

∃N 1 ∈N:∀n∈N:n>N 1 :|L1 −xn|<ϵ 

∃N 2 ∈N:∀n∈N:n>N 2 :| L2 −xn|<ϵ   �if N=max{N 1 ,N 2 }, xn would be arbitrary close to L1 and arbitrary close to L2 at the same, it is impossible-this is the contradiction (Unless L1 =L2)

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Index

FAQ

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Limit of Sums

Theorem

Proof

Index

FAQ

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Limit of Sums

Proof

By the Triangle Inequality

Index

FAQ

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Limits of Products

The same argument as for sums can be used to prove the following result.

Theorem

Remark

Examples

Index

FAQ

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Squeeze Theorem for Sequences

Theorem

Proof

Index

FAQ

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Using the Squeeze/Pinching Theorem

Example

Solution

This is difficult to compute using the standard methods because n! is defined only if n is a natural number.

So the values of the sequence in question are not given by an elementary function to which we could apply tricks like L’Hospital’s Rule.

Here each term k/n < 1.

Index

FAQ

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Using the Squeeze Theorem

Problem

Solution

Index

FAQ

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Monotonous Sequences

Definition

The sequence (a1,a2,a3,…) is decreasing if an+1 ≤ an for all n.

A sequence (a1,a2,a3,…) is increasing if an ≤ an+1 for all n.

The sequence (a1,a2,a3,…) is monotonous if it is either increasing or decreasing.

Theorem

The sequence (a1,a2,a3,…) is bounded if there are numbers M and m such that m ≤ an ≤ M for all n.

A bounded monotonous sequence always has a finite limit.

Observe that it suffices to show that the theorem for increasing sequences (an) since if (an) is decreasing, then consider the increasing sequence (-an).

Index

FAQ

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Monotonous Sequences

Theorem

A bounded monotonous sequence always has a finite limit.

Proof

Let (a1,a2,a3,…) be an increasing bounded sequence.

Then the set {a1,a2,a3,…} is bounded from the above.

By the fact that the set of real numbers is complete, � s=sup {a1,a2,a3,…} �is finite.

Claim

Index

FAQ

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Monotonous Sequences

Theorem

A bounded monotonous sequence always has a finite limit.

Proof

Let (a1,a2,a3,…) be an increasing bounded sequence.

Let s=sup {a1,a2,a3,…}.

Claim

Proof of the Claim

Index

FAQ