Q. 84 - Calculus

Question

Prove that if $\underset{k\to \infty }{\lim }{a}_k=L,$then $\underset{k\to \infty }{\lim }{a}_{k+1}=L$

Step-by-Step Solution

Answer

Hence proved that $\underset{k\to \infty }{\lim }{a}_{k+1}=L$

Step 1. Given information

The given sequence $\underset{k\to \infty }{\lim }{a}_k=L$

Step 2. The strategy to prove that lim k → ∞ a k + 1 = L .

Use the defination of convergence of sequence$\left\{a_k\right\}$

The sequence $\left\{a_k\right\}$ is convergent and convergers to L.

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

$\left|a_k-L\right|<\epsilon for k\ge N$

The result $\left|a_k-L\right|<\epsilon$is true for all $k\ge N$

Since the following inequality holds

$k+1>k$

Therefore

$k+1>N (Because k\ge N)$

Therefore,there is a positive integer N such that

$\left|a_{k+1}-L\right|<\epsilon for k+1>N$

Thus,$\underset{k\to \infty }{\lim }{a}_{k+1}=L$