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Chapter 3: Problem 3

Show that if \(X\) and \(Y\) are sequences such that \(X\) and \(X+Y\) are convergent, then \(Y\) is convergent.

Short Answer

Expert verified
To prove that sequence \(Y\) is convergent, express it in terms of convergent sequences \(X\) and \(X+Y\) as \(Y = (X + Y) - X\). Apply the algebraic property of limits, \( \lim_{n \rightarrow \infty} Y = \lim_{n \rightarrow \infty} (X + Y) - \lim_{n \rightarrow \infty} X\), which becomes \( \lim_{n \rightarrow \infty} Y = b - a\). So \(Y\) is convergent because \(b - a\) is a real number, establishing a finite limit for sequence \(Y\).

Step by step solution

01

Setup

Assume that \(X\) and \(X+Y\) are both convergent sequences. Let the limit of sequence \(X\) be \(a\) and the limit of sequence \(X + Y\) be \(b\). So we can write that \[\lim_{n \rightarrow \infty} X = a\] \[\lim_{n \rightarrow \infty} (X+Y) = b\]
02

Express Y sequence

We can express sequence \(Y\) in terms of \(X\) and \(X + Y\). This can be done by subtracting sequence \(X\) from sequence \(X+Y\): \[Y = (X + Y) - X\]
03

Apply properties of limits

As both \(X\) and \(X + Y\) are convergent, we can subtract them according to the algebraic properties of limits: \[\lim_{n \rightarrow \infty} Y = \lim_{n \rightarrow \infty} (X + Y) - \lim_{n \rightarrow \infty} X\] Substituting the limits we stated in step 1: \[\lim_{n \rightarrow \infty} Y = b - a\]
04

Conclude

Since \(b - a\) is a real number, sequence \(Y\) has a finite limit and hence it's convergent.

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Most popular questions from this chapter

The sequence \(\left(x_{n}\right)\) is defined by che following formulas for the \(n\) th term. Write the furst five terms in each case: (a) \(x_{n}:=1+(-1)^{n}\), (b) \(x_{n}:=(-1)^{n} / n\), (c) \(x_{n}:=\frac{1}{n(n+1)}\), (d) \(x:=\frac{1}{n^{2}+2}\).

Show that the following sequences are divergent. (a) \(\left(1-(-1)^{n}+1 / n\right)\), (b) \((\sin n \pi / 4)\).

Show that \(\lim \left(1 / 3^{n}\right)=0\).

Show that \(\lim \left(\frac{1}{n}-\frac{1}{n+1}\right)=0\).

Let \(X=\left(x_{n}\right)\) and \(Y=\left(y_{n}\right)\) be given sequences, and let the "shuffled" sequence \(Z=\left(z_{n}\right)\) be defined by \(z_{1}:=x_{1}, z_{2}:=y_{1}, \cdots, z_{2 n-1}:=x_{n}, z_{2 n}:=y_{n}, \cdots .\) Show that \(Z\) is convergent if and only if both \(X\) and \(Y\) are convergent and \(\lim X=\lim Y\).
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