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Chapter 3: Problem 11

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

Short Answer

Expert verified
The limit of the sequence \(\frac{1}{n}-\frac{1}{n+1}\) is \(0\).

Step by step solution

01

Combine the two fractions

First, we need to combine our two fractions. The common denominator of fractions \(\frac{1}{n}\) and \(\frac{1}{n+1}\) is \(n*(n+1)\). Applying this common denominator to both fractions we get \(\frac{n+1-n}{n*(n+1)}\). This simplifies to \(\frac{1}{n*(n+1)}\). So, our sequence becomes \(\frac{1}{n*(n+1)}\).
02

Find the limit

After simplifying the sequence, we can find its limit directly. If we substitute \(n\) with an \(鈭瀄) into \(\frac{1}{n*(n+1)}\) we get \(0\), since division by infinity equals zero. So, the limit of the sequence is \(0\).

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