Question

Find the limit (if exist) of the sequence {Xn]nen of real numbers defined by xn = 1 +1/2+1/(2^2)+.....+2(2^(n-1)),for all n€N
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Answer 1

Answer:

Step 1

Show that

0<xn≤3(1)

for n=1,2,⋯.

Proof

Obviously, (1) holds for n=1. Assume that (1) holds for n=k, namely, 0<xk≤3, then

0<14<xk+1=14−xk≤1≤3,

which implies (1) also holds for n=k+1. Thus, by mathematical induction, (1) holds for all n=1,2,⋯.

Step 2

Show that

xn+1<xn(2)

for n=1,2,⋯.

Proof

Notice that x1=3,x2=1. Hence x2<x1, which implies that (2) holds for n=1. Assume that (2) holds for n=k, namely, xk+1<xk, then

xk+2=14−xk+1<14−xk=xk+1,

which implies (2) also holds for n=k+1. Thus, by mathematical induction, (2) holds for all n=1,2,⋯.

Combining the two steps, by monotone convergence theorem, we may claim xn has a limit, which could be denoted as x. Thus, taking the simultaneous limits of both sides of the recursion formula, we have

x=14−x.

Hence, x=2±3–√. But xn≤3, hence x≤3, which implies that 2+3–√>3 does not satisfy the requirement. As a result,

x=2−3–√,

which is desired limit.