Q17P Here are some other ways of obta... [FREE SOLUTION]

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Chapter 4: Q17P (page 199)

Here are some other ways of obtaining the formula in Example 2.
a) Combine the two fractions to get $(2 n + 1) / [n^{2} (n + 1)]$ . Then note that for large n, $2 n + 1 鈮� 2 n$ and $n + 1 鈮� n$ .
b) Factor the expression as $\frac{1}{n^{2}} (1 - \frac{1}{{(1 + \frac{1}{n + 1})}^{2}})$ , expand ${(1 + \frac{1}{n + 1})}^{- 2}$ by binomial series to two terms, and then simplify.

Short Answer

Expert verified

(a). The solution is $\frac{1}{{(n + 1)}^{2}} - \frac{n^{2}}{n^{2} {(n + 1)}^{2}} 鈮� \frac{2 n}{n^{3}}$ .

(b). The solution is $\frac{1}{{(n + 1)}^{2}} - \frac{n^{2}}{n^{2} {(n + 1)}^{2}} = \frac{2 n}{n^{3}}$ .

Step by step solution

Part (a)

Given fractions in example 2 is, $\frac{1}{n^{2}} - \frac{1}{{(n + 1)}^{2}}$ .

Combine the two fractions to find the limit as approaches infinity.

$\frac{1}{n^{2}} - \frac{1}{{(n + 1)}^{2}} = \frac{{(n + 1)}^{2} - n^{2}}{n^{2} {(n + 1)}^{2}} = \frac{n^{2} + 2 n + 1 - n^{2}}{n^{2} {(n + 1)}^{2}} = \frac{2 n + 1}{n^{2} {(n + 1)}^{2}}$

For very large n, then $2 n + 1 鈮� 2 n a n d n + 1 鈮� n .$

$\frac{2 n + 1}{n^{2} {(n + 1)}^{2}} 鈮� \frac{2 n}{n^{3}} \frac{1}{{(n + 1)}^{2}} - \frac{n^{2}}{n^{2} {(n + 1)}^{2}} 鈮� \frac{2 n}{n^{3}}$

Part (b)

Consider the well factoring form $\frac{1}{n^{2}}$ the left-hand side of $\frac{1}{n^{2}} - \frac{1}{{(n + 1)}^{2}} 鈮� \frac{2}{n^{3}}$ then use the binomial theorem on ${(1 + \frac{1}{n})}^{- 2}$ .

$\frac{1}{n^{2}} + \frac{1}{{(n + 1)}^{2}} = \frac{1}{n^{2}} (1 - \frac{1}{{(1 + \frac{1}{n})}^{2}}) = \frac{1}{n^{2}} (1 - {(1 + \frac{1}{n})}^{- 2})$

Use the binomial theorem on ${(1 + \frac{1}{n})}^{- 2}$ (two terms).

${(a + b)}^{n} = a^{n} + n a^{n - 1} + \frac{n (n + 1)}{2!} a^{n - 2} b^{2} + . . . + b^{n}$

Then, ${(1 + \frac{1}{n + 1})}^{- 2} 鈮� 1 + (- 2) 脳 \frac{1}{n}$ . Which implies that,

$\frac{1}{n^{2}} (1 - \frac{1}{{(1 + \frac{1}{n + 1})}^{2}}) 鈮� \frac{1}{n^{2}} (1 - (1 - \frac{2}{n}))$

Then, we have

$\frac{1}{n^{2}} - \frac{1}{{(n + 1)}^{2}} = \frac{1}{n^{2}} 脳 \frac{2}{n} = \frac{2}{n^{3}}$

Which is the required solution.

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Short Answer

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Part (a)

Part (b)

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91影视

Here are some other ways of obtaining the formula in Example 2.a) Combine the two fractions to get(2n+1)/[n2(n+1)]. Then note that for large n,2n+1鈮�2nandn+1鈮�n.b) Factor the expression as 1n2(1-11+1n+12), expand(1+1n+1)-2 by binomial series to two terms, and then simplify.

Short Answer

Step by step solution

Part (a)

Part (b)

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Most popular questions from this chapter

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Mechanics and Materials

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