Answer to Question #196741 in Real Analysis for Saurabh Sharma

Question

Consider the sequence {xn}, which is defined by

x1 = 1, xn+1 = xn +

1/

x1 + x2 + · · · + xn

, n ∈ N.

Does {xn} converge? Justify your answer.

Answer 1

$x_1=1,x_{n+1}=x_n+1, x_1+x_2+....+x_n\in N$

Given sequence is-

$x_{n+1}=x_n+1$

$\Rightarrow x_2=x_1+1=2 \\ \Rightarrow x_3=x_2+1=2+1=3$

We get the sequence -

$1,2,3,4....,n$

The $n^{th}$ term is-

$a_n=1+(n-1)1=n$

Using ratio test-

$lim_{n\to \infty}\dfrac{a_{n+1}}{a_n}=lim_{n\to \infty}\dfrac{n+1}{n}$

$=lim_{n\to \infty} 1+\dfrac{1}{n}\\[9pt]=1$

Hence The given sequence $x_n$ converges to 1.