Chapter 7: Q. 84 (page 605)

Prove that if $\lim_{k \to \infty} a_{k} = L,$ then localid="1649337757642" $\lim_{k \to \infty} a_{k + 1} = L$

Short Answer

Expert verified

Hence proved that $\lim_{k \to \infty} a_{k + 1} = L$

Step by step solution

Step 1. Given information

The given sequence $\lim_{k \to \infty} a_{k} = L$

Step 2. The strategy to prove that limk→∞ak+1=L.

Use the defination of convergence of sequence $\{a_{k}\}$

The sequence $\{a_{k}\}$ is convergent and convergers to L.

By the defination of convergence,for $ε > 0$ there is a positive integer N,such that

$|a_{k} - L| < ε f o r k \geq N$

The result $|a_{k} - L| < ε$ is true for all $k \geq N$

Since the following inequality holds

$k + 1 > k$

Therefore

$k + 1 > N (B e c a u s e k \geq N)$

Therefore,there is a positive integer Nsuch that

$|a_{k + 1} - L| < ε f o r k + 1 > N$

Thus, $\lim_{k \to \infty} a_{k + 1} = L$

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Prove that if $\lim_{k \to \infty} a_{k} = L,$ then localid="1649337757642" $\lim_{k \to \infty} a_{k + 1} = L$

Short Answer

Step by step solution

Step 1. Given information

Step 2. The strategy to prove that limk→∞ak+1=L.

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Prove that if limk→∞ak=L,then localid="1649337757642" limk→∞ak+1=L

Short Answer

Step by step solution

Step 1. Given information

Step 2. The strategy to prove that limk→∞ak+1=L.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Prove that if $\lim_{k \to \infty} a_{k} = L,$ then localid="1649337757642" $\lim_{k \to \infty} a_{k + 1} = L$