Answer to Question #196741 in Real Analysis for Saurabh Sharma

Question #196741

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.


1
Expert's answer
2022-01-31T17:37:32-0500

"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\n\n\n\\\\\n\\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.


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