Q. 11 Show that聽鈭慿=081k2+1聽is聽equ... [FREE SOLUTION]

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Chapter 4: Q. 11 (page 325)

Show that ${鈭�}_{k = 0}^{8} \frac{1}{k^{2} + 1} is equal to 2 {鈭�}_{k = 0}^{8} \frac{1}{2 k^{2} + 2}$ by writing out the terms in each sum.

Short Answer

Expert verified

It is shown that ${鈭�}_{k = 0}^{8} \frac{1}{k^{2} + 1} = 2 {鈭�}_{k = 0}^{8} \frac{1}{2 k^{2} + 2}$

Step by step solution

Step 1. Given Information

The given summations are ${鈭�}_{k = 0}^{8} \frac{1}{k^{2} + 1} a n d 2 {鈭�}_{k = 0}^{8} \frac{1}{2 k^{2} + 2}$

Step 2. Explanation

The first sum can be written as follows,

${鈭�}_{k = 0}^{8} \frac{1}{k^{2} + 1} = 1 + \frac{1}{1 + 1} + \frac{1}{4 + 1} + \frac{1}{9 + 1} + \frac{1}{16 + 1} + \frac{1}{25 + 1} + \frac{1}{36 + 1} + \frac{1}{49 + 1} + \frac{1}{64 + 1} = 1 + \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} + \frac{1}{26} + \frac{1}{37} + \frac{1}{50} + \frac{1}{65}$

and the second sum can be written as,

$2 {鈭�}_{k = 0}^{8} \frac{1}{2 k^{2} + 2} = 2 (\frac{1}{0 + 2} + \frac{1}{2 {(1)}^{2} + 2} + \frac{1}{2 {(2)}^{2} + 2} + \frac{1}{2 {(3)}^{2} + 2} + \frac{1}{2 {(4)}^{2} + 2} + \frac{1}{2 {(5)}^{2} + 2} + \frac{1}{2 {(6)}^{2} + 2} + \frac{1}{2 {(7)}^{2} + 2} + \frac{1}{2 {(8)}^{2} + 2}) = 2 (\frac{1}{2} + \frac{1}{4} + \frac{1}{10} + \frac{1}{20} + \frac{1}{34} + \frac{1}{52} + \frac{1}{74} + \frac{1}{100} + \frac{1}{130}) = 1 + \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} + \frac{1}{26} + \frac{1}{37} + \frac{1}{50} + \frac{1}{65}$

Hence, we can say that both the summations are equal

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91影视

Show that ${鈭�}_{k = 0}^{8} \frac{1}{k^{2} + 1} is equal to 2 {鈭�}_{k = 0}^{8} \frac{1}{2 k^{2} + 2}$ by writing out the terms in each sum.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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Show that鈭�k=081k2+1isequalto2鈭�k=0812k2+2by writing out the terms in each sum.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Explanation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

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Geometry

Study anywhere. Anytime. Across all devices.

Show that ${鈭�}_{k = 0}^{8} \frac{1}{k^{2} + 1} is equal to 2 {鈭�}_{k = 0}^{8} \frac{1}{2 k^{2} + 2}$ by writing out the terms in each sum.